9 14

9_13

9_15

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 14's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 14 at Knotilus!

Knot presentations

Planar diagram presentation	X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X13,18,14,1 X9,15,10,14 X7,17,8,16 X15,9,16,8 X17,7,18,6
Gauss code	-1, 4, -3, 1, -2, 9, -7, 8, -6, 3, -4, 2, -5, 6, -8, 7, -9, 5
Dowker-Thistlethwaite code	4 10 12 16 14 2 18 8 6
Conway Notation	[41112]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 10, width is 5, Braid index is 5		[{11, 3}, {2, 9}, {10, 4}, {3, 5}, {9, 11}, {4, 1}, {6, 2}, {5, 7}, {8, 6}, {7, 10}, {1, 8}]

[edit Notes on presentations of 9 14]

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.org/wiki/KnotTheory.

In[3]:=

K = Knot["9 14"];

In[4]:=

PD[K]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

X1425 X5,12,6,13 X3,11,4,10 X11,3,12,2 X13,18,14,1 X9,15,10,14 X7,17,8,16 X15,9,16,8 X17,7,18,6

In[5]:=

GaussCode[K]

Out[5]=

-1, 4, -3, 1, -2, 9, -7, 8, -6, 3, -4, 2, -5, 6, -8, 7, -9, 5

In[6]:=

DTCode[K]

Out[6]=

4 10 12 16 14 2 18 8 6

(The path below may be different on your system)

In[7]:=

AppendTo[$Path, "C:/bin/LinKnot/"];

In[8]:=

ConwayNotation[K]

Out[8]=

[41112]

In[9]:=

br = BR[K]

KnotTheory::credits: The minimum braids representing the knots with up to 10 crossings were provided by Thomas Gittings. See arXiv:math.GT/0401051.

Out[9]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \textrm{BR}(5,\{1,1,2,-1,-3,2,-3,4,-3,4\})}

In[10]:=

{First[br], Crossings[br], BraidIndex[K]}

KnotTheory::credits: The braid index data known to KnotTheory` is taken from Charles Livingston's http://www.indiana.edu/~knotinfo/.

KnotTheory::loading: Loading precomputed data in IndianaData`.

Out[10]=

{ 5, 10, 5 }

In[11]:=

Show[BraidPlot[br]]

Out[11]=

-Graphics-

In[12]:=

Show[DrawMorseLink[K]]

KnotTheory::credits: "MorseLink was added to KnotTheory` by Siddarth Sankaran at the University of Toronto in the summer of 2005."

KnotTheory::credits: "DrawMorseLink was written by Siddarth Sankaran at the University of Toronto in the summer of 2005."

Out[12]=

-Graphics-

In[13]:=

ap = ArcPresentation[K]

Out[13]=

ArcPresentation[{11, 3}, {2, 9}, {10, 4}, {3, 5}, {9, 11}, {4, 1}, {6, 2}, {5, 7}, {8, 6}, {7, 10}, {1, 8}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{4,7\}} Nakanishi index 1 Maximal Thurston-Bennequin number [-4][-7] Hyperbolic Volume 8.95499 A-Polynomial See Data:9 14/A-polynomial

[edit Notes for 9 14's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} Topological 4 genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 1} Concordance genus Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2} Rasmussen s-Invariant 0

[edit Notes for 9 14's four dimensional invariants]

Polynomial invariants

Alexander polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 t^2-9 t+15-9 t^{-1} +2 t^{-2} }
Conway polynomial	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 z^4-z^2+1}
2nd Alexander ideal (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}
Determinant and Signature	{ 37, 0 }
Jones polynomial	$q^{6}-2q^{5}+3q^{4}-5q^{3}+6q^{2}-6q+6-4q^{-1}+3q^{-2}-q^{-3}$
HOMFLY-PT polynomial (db, data sources)	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^4 a^{-2} +z^4-a^2 z^2+z^2 a^{-2} -2 z^2 a^{-4} +z^2+ a^{-2} -2 a^{-4} + a^{-6} +1}
Kauffman polynomial (db, data sources)	$z^{8}a^{-2}+z^{8}a^{-4}+3z^{7}a^{-1}+5z^{7}a^{-3}+2z^{7}a^{-5}+3z^{6}a^{-2}+z^{6}a^{-6}+4z^{6}+4az^{5}-4z^{5}a^{-1}-16z^{5}a^{-3}-8z^{5}a^{-5}+3a^{2}z^{4}-12z^{4}a^{-2}-9z^{4}a^{-4}-4z^{4}a^{-6}-4z^{4}+a^{3}z^{3}-3az^{3}+2z^{3}a^{-1}+15z^{3}a^{-3}+9z^{3}a^{-5}-2a^{2}z^{2}+8z^{2}a^{-2}+10z^{2}a^{-4}+4z^{2}a^{-6}-2za^{-1}-5za^{-3}-3za^{-5}-a^{-2}-2a^{-4}-a^{-6}+1$
The A2 invariant	$-q^{10}+q^{8}+q^{6}-q^{4}+2q^{2}+q^{-2}+q^{-4}+q^{-8}-2q^{-10}-q^{-12}-q^{-16}+q^{-18}+q^{-20}$
The G2 invariant	$q^{52}-2q^{50}+3q^{48}-4q^{46}+q^{44}-3q^{40}+8q^{38}-10q^{36}+11q^{34}-8q^{32}+2q^{30}+4q^{28}-10q^{26}+16q^{24}-16q^{22}+14q^{20}-8q^{18}-q^{16}+11q^{14}-15q^{12}+17q^{10}-12q^{8}+3q^{6}+6q^{4}-10q^{2}+10-2q^{-2}-6q^{-4}+15q^{-6}-16q^{-8}+9q^{-10}+5q^{-12}-19q^{-14}+30q^{-16}-28q^{-18}+18q^{-20}-16q^{-24}+28q^{-26}-30q^{-28}+23q^{-30}-10q^{-32}-6q^{-34}+16q^{-36}-19q^{-38}+15q^{-40}-5q^{-42}-7q^{-44}+11q^{-46}-13q^{-48}+5q^{-50}+5q^{-52}-16q^{-54}+21q^{-56}-19q^{-58}+6q^{-60}+8q^{-62}-20q^{-64}+25q^{-66}-21q^{-68}+11q^{-70}+q^{-72}-10q^{-74}+16q^{-76}-14q^{-78}+10q^{-80}-2q^{-82}-2q^{-84}+3q^{-86}-4q^{-88}+3q^{-90}-q^{-92}+q^{-94}$

Further Quantum Invariants

Further quantum knot invariants for 9_14.

A1 Invariants.

Weight	Invariant
1	$-q^{7}+2q^{5}-q^{3}+2q+q^{-5}-2q^{-7}+q^{-9}-q^{-11}+q^{-13}$
2	$q^{20}-2q^{18}-q^{16}+4q^{14}-4q^{12}+7q^{8}-5q^{6}-q^{4}+7q^{2}-3-3q^{-2}+3q^{-4}+2q^{-6}-3q^{-8}-2q^{-10}+5q^{-12}-q^{-14}-6q^{-16}+5q^{-18}+2q^{-20}-6q^{-22}+4q^{-24}+4q^{-26}-5q^{-28}+3q^{-32}-2q^{-34}-q^{-36}+q^{-38}$
3	$-q^{39}+2q^{37}+q^{35}-2q^{33}-2q^{31}+q^{29}+3q^{27}-5q^{25}+6q^{21}-10q^{17}+3q^{15}+14q^{13}-q^{11}-17q^{9}+19q^{5}+5q^{3}-16q-9q^{-1}+10q^{-3}+11q^{-5}-2q^{-7}-14q^{-9}-3q^{-11}+12q^{-13}+10q^{-15}-11q^{-17}-13q^{-19}+8q^{-21}+16q^{-23}-6q^{-25}-16q^{-27}+2q^{-29}+18q^{-31}+2q^{-33}-17q^{-35}-6q^{-37}+16q^{-39}+12q^{-41}-13q^{-43}-15q^{-45}+5q^{-47}+16q^{-49}-q^{-51}-14q^{-53}-4q^{-55}+10q^{-57}+7q^{-59}-5q^{-61}-6q^{-63}+2q^{-65}+4q^{-67}-2q^{-71}-q^{-73}+q^{-75}$
4	$q^{64}-2q^{62}-q^{60}+2q^{58}+5q^{54}-4q^{52}-2q^{50}-6q^{46}+11q^{44}-2q^{42}+3q^{40}-21q^{36}+4q^{34}+6q^{32}+26q^{30}+10q^{28}-43q^{26}-22q^{24}+4q^{22}+56q^{20}+40q^{18}-48q^{16}-56q^{14}-24q^{12}+60q^{10}+70q^{8}-16q^{6}-54q^{4}-55q^{2}+21+63q^{-2}+24q^{-4}-14q^{-6}-52q^{-8}-26q^{-10}+19q^{-12}+41q^{-14}+30q^{-16}-25q^{-18}-48q^{-20}-17q^{-22}+41q^{-24}+47q^{-26}-3q^{-28}-53q^{-30}-36q^{-32}+39q^{-34}+51q^{-36}+8q^{-38}-57q^{-40}-48q^{-42}+34q^{-44}+54q^{-46}+29q^{-48}-47q^{-50}-64q^{-52}+8q^{-54}+45q^{-56}+56q^{-58}-14q^{-60}-63q^{-62}-29q^{-64}+8q^{-66}+62q^{-68}+30q^{-70}-28q^{-72}-41q^{-74}-34q^{-76}+32q^{-78}+44q^{-80}+15q^{-82}-17q^{-84}-47q^{-86}-6q^{-88}+21q^{-90}+27q^{-92}+12q^{-94}-25q^{-96}-17q^{-98}-4q^{-100}+12q^{-102}+16q^{-104}-4q^{-106}-6q^{-108}-6q^{-110}+6q^{-114}+q^{-116}-2q^{-120}-q^{-122}+q^{-124}$
5	$-q^{95}+2q^{93}+q^{91}-2q^{89}-3q^{85}-2q^{83}+3q^{81}+7q^{79}+3q^{77}-q^{75}-8q^{73}-12q^{71}-4q^{69}+13q^{67}+26q^{65}+10q^{63}-15q^{61}-36q^{59}-38q^{57}+9q^{55}+66q^{53}+62q^{51}+q^{49}-80q^{47}-112q^{45}-33q^{43}+111q^{41}+167q^{39}+71q^{37}-114q^{35}-226q^{33}-137q^{31}+104q^{29}+277q^{27}+209q^{25}-60q^{23}-298q^{21}-276q^{19}-8q^{17}+276q^{15}+324q^{13}+90q^{11}-216q^{9}-326q^{7}-162q^{5}+119q^{3}+282q+208q^{-1}-17q^{-3}-202q^{-5}-216q^{-7}-66q^{-9}+103q^{-11}+184q^{-13}+132q^{-15}-13q^{-17}-142q^{-19}-158q^{-21}-53q^{-23}+90q^{-25}+172q^{-27}+104q^{-29}-65q^{-31}-169q^{-33}-121q^{-35}+46q^{-37}+177q^{-39}+134q^{-41}-48q^{-43}-189q^{-45}-148q^{-47}+46q^{-49}+209q^{-51}+168q^{-53}-43q^{-55}-223q^{-57}-200q^{-59}+18q^{-61}+233q^{-63}+234q^{-65}+20q^{-67}-213q^{-69}-260q^{-71}-80q^{-73}+172q^{-75}+272q^{-77}+137q^{-79}-105q^{-81}-250q^{-83}-190q^{-85}+19q^{-87}+202q^{-89}+217q^{-91}+59q^{-93}-122q^{-95}-199q^{-97}-131q^{-99}+32q^{-101}+157q^{-103}+159q^{-105}+50q^{-107}-79q^{-109}-149q^{-111}-112q^{-113}+3q^{-115}+104q^{-117}+127q^{-119}+61q^{-121}-42q^{-123}-107q^{-125}-95q^{-127}-16q^{-129}+65q^{-131}+93q^{-133}+51q^{-135}-18q^{-137}-65q^{-139}-62q^{-141}-15q^{-143}+34q^{-145}+51q^{-147}+27q^{-149}-7q^{-151}-29q^{-153}-28q^{-155}-6q^{-157}+14q^{-159}+17q^{-161}+7q^{-163}-2q^{-165}-8q^{-167}-8q^{-169}+4q^{-173}+3q^{-175}+q^{-177}-2q^{-181}-q^{-183}+q^{-185}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{10}+q^{8}+q^{6}-q^{4}+2q^{2}+q^{-2}+q^{-4}+q^{-8}-2q^{-10}-q^{-12}-q^{-16}+q^{-18}+q^{-20}$
1,1	$q^{28}-4q^{26}+8q^{24}-12q^{22}+18q^{20}-28q^{18}+34q^{16}-38q^{14}+43q^{12}-48q^{10}+50q^{8}-44q^{6}+46q^{4}-32q^{2}+22-24q^{-4}+50q^{-6}-80q^{-8}+102q^{-10}-118q^{-12}+126q^{-14}-126q^{-16}+108q^{-18}-91q^{-20}+62q^{-22}-30q^{-24}+34q^{-28}-50q^{-30}+70q^{-32}-76q^{-34}+71q^{-36}-62q^{-38}+48q^{-40}-36q^{-42}+21q^{-44}-12q^{-46}+6q^{-48}-2q^{-50}+q^{-52}$
2,0	$q^{26}-q^{24}-2q^{22}+q^{20}+2q^{18}-q^{16}-4q^{14}+3q^{12}+5q^{10}-3q^{8}-2q^{6}+6q^{4}+4q^{2}-2-2q^{-2}+2q^{-4}-q^{-8}+q^{-10}-3q^{-14}+2q^{-16}+q^{-18}-5q^{-20}-3q^{-22}+2q^{-24}+3q^{-26}-2q^{-28}+5q^{-32}+4q^{-34}-2q^{-36}-2q^{-38}+q^{-40}+q^{-42}-2q^{-44}-3q^{-46}+q^{-50}+q^{-52}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{22}-2q^{20}-q^{18}+4q^{16}-3q^{14}-2q^{12}+6q^{10}-2q^{8}-4q^{6}+7q^{4}+q^{2}-1+5q^{-2}+2q^{-4}-2q^{-6}-3q^{-8}-6q^{-14}+2q^{-16}+5q^{-18}-4q^{-20}+2q^{-22}+4q^{-24}-4q^{-26}+2q^{-30}-3q^{-32}+q^{-34}+q^{-36}-q^{-38}+q^{-40}$
1,0,0	$-q^{13}+q^{11}+q^{7}-q^{5}+2q^{3}+q^{-1}+q^{-3}+q^{-5}+q^{-7}+q^{-11}-2q^{-13}-q^{-15}-2q^{-17}-q^{-21}+q^{-23}+q^{-25}+q^{-27}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{22}+2q^{20}-3q^{18}+4q^{16}-5q^{14}+6q^{12}-6q^{10}+6q^{8}-4q^{6}+3q^{4}+q^{2}-3+7q^{-2}-8q^{-4}+12q^{-6}-11q^{-8}+12q^{-10}-10q^{-12}+8q^{-14}-6q^{-16}+q^{-18}-4q^{-22}+4q^{-24}-6q^{-26}+6q^{-28}-6q^{-30}+5q^{-32}-3q^{-34}+3q^{-36}-q^{-38}+q^{-40}$
1,0	$q^{36}-2q^{32}-2q^{30}+q^{28}+4q^{26}+q^{24}-4q^{22}-4q^{20}+2q^{18}+6q^{16}+2q^{14}-5q^{12}-4q^{10}+3q^{8}+7q^{6}-4q^{2}+6q^{-2}+3q^{-4}-3q^{-6}-4q^{-8}+2q^{-10}+3q^{-12}-2q^{-14}-5q^{-16}+3q^{-20}-q^{-22}-5q^{-24}-q^{-26}+6q^{-28}+4q^{-30}-3q^{-32}-6q^{-34}+2q^{-36}+7q^{-38}+3q^{-40}-5q^{-42}-5q^{-44}+2q^{-46}+5q^{-48}-4q^{-52}-2q^{-54}+2q^{-56}+2q^{-58}-q^{-60}-q^{-62}+q^{-66}$

G2 Invariants.

Weight

Invariant

1,0

$q^{52}-2q^{50}+3q^{48}-4q^{46}+q^{44}-3q^{40}+8q^{38}-10q^{36}+11q^{34}-8q^{32}+2q^{30}+4q^{28}-10q^{26}+16q^{24}-16q^{22}+14q^{20}-8q^{18}-q^{16}+11q^{14}-15q^{12}+17q^{10}-12q^{8}+3q^{6}+6q^{4}-10q^{2}+10-2q^{-2}-6q^{-4}+15q^{-6}-16q^{-8}+9q^{-10}+5q^{-12}-19q^{-14}+30q^{-16}-28q^{-18}+18q^{-20}-16q^{-24}+28q^{-26}-30q^{-28}+23q^{-30}-10q^{-32}-6q^{-34}+16q^{-36}-19q^{-38}+15q^{-40}-5q^{-42}-7q^{-44}+11q^{-46}-13q^{-48}+5q^{-50}+5q^{-52}-16q^{-54}+21q^{-56}-19q^{-58}+6q^{-60}+8q^{-62}-20q^{-64}+25q^{-66}-21q^{-68}+11q^{-70}+q^{-72}-10q^{-74}+16q^{-76}-14q^{-78}+10q^{-80}-2q^{-82}-2q^{-84}+3q^{-86}-4q^{-88}+3q^{-90}-q^{-92}+q^{-94}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 14"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 t^2-9 t+15-9 t^{-1} +2 t^{-2} }

In[5]:=

Conway[K][z]

Out[5]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle 2 z^4-z^2+1}

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \{1\}}

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 37, 0 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$q^{6}-2q^{5}+3q^{4}-5q^{3}+6q^{2}-6q+6-4q^{-1}+3q^{-2}-q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle z^4 a^{-2} +z^4-a^2 z^2+z^2 a^{-2} -2 z^2 a^{-4} +z^2+ a^{-2} -2 a^{-4} + a^{-6} +1}

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$z^{8}a^{-2}+z^{8}a^{-4}+3z^{7}a^{-1}+5z^{7}a^{-3}+2z^{7}a^{-5}+3z^{6}a^{-2}+z^{6}a^{-6}+4z^{6}+4az^{5}-4z^{5}a^{-1}-16z^{5}a^{-3}-8z^{5}a^{-5}+3a^{2}z^{4}-12z^{4}a^{-2}-9z^{4}a^{-4}-4z^{4}a^{-6}-4z^{4}+a^{3}z^{3}-3az^{3}+2z^{3}a^{-1}+15z^{3}a^{-3}+9z^{3}a^{-5}-2a^{2}z^{2}+8z^{2}a^{-2}+10z^{2}a^{-4}+4z^{2}a^{-6}-2za^{-1}-5za^{-3}-3za^{-5}-a^{-2}-2a^{-4}-a^{-6}+1$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n53,}

Computer Talk

The above data is available with the Mathematica package KnotTheory`. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of May 31, 2006, 14:15:20.091. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["9 14"];

In[4]:=

{A = Alexander[K][t], J = Jones[K][q]}

KnotTheory::loading: Loading precomputed data in PD4Knots`.

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[4]=

{ $2t^{2}-9t+15-9t^{-1}+2t^{-2}$, $q^{6}-2q^{5}+3q^{4}-5q^{3}+6q^{2}-6q+6-4q^{-1}+3q^{-2}-q^{-3}$ }

In[5]:=

DeleteCases[Select[AllKnots[], (A === Alexander[#][t]) &], K]

KnotTheory::loading: Loading precomputed data in DTCode4KnotsTo11`.

KnotTheory::credits: The GaussCode to PD conversion was written by Siddarth Sankaran at the University of Toronto in the summer of 2005.

Out[5]=

{}

In[6]:=

DeleteCases[ Select[ AllKnots[], (J === Jones[#][q] || (J /. q -> 1/q) === Jones[#][q]) & ], K ]

KnotTheory::loading: Loading precomputed data in Jones4Knots11`.

Out[6]=

{K11n53,}

Vassiliev invariants

V2 and V3:

(-1, -2)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $-4$ $-16$ $8$ $-{\frac {110}{3}}$ $-{\frac {34}{3}}$ $64$ ${\frac {32}{3}}$ ${\frac {128}{3}}$ $-48$ $-{\frac {32}{3}}$ $128$ ${\frac {440}{3}}$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{136}{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{10529}{30}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{1502}{15}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{4978}{45}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle -\frac{833}{18}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \frac{449}{30}}

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle t^rq^j} are shown, along with their alternating sums Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \chi} (fixed Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j} , alternation over Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r} ). The squares with yellow highlighting are those on the "critical diagonals", where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s+1} or Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle j-2r=s-1} , where Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle s=} 0 is the signature of 9 14. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 4 5 6 χ 13                   1 1 11                 1   -1 9               2 1   1 7             3 1     -2 5           3 2       1 3         3 3         0 1       3 3           0 -1     2 4             2 -3   1 2               -1 -5   2                 2 -7 1                   -1

Integral Khovanov Homology (db, data source)

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle \dim{\mathcal G}_{2r+i}\operatorname{KH}^r_{\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle i=1} $r=-3$ ${\mathbb {Z} }$ Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=-1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=0} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{4}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=1} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=3} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}\oplus{\mathbb Z}_2^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{3}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=4} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}^{2}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=5} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}\oplus{\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle r=6} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}_2} Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle {\mathbb Z}}

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n+1} -dimensional representation of Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle sl(2)} )

Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle n}	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle J_n}
2	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{18}-2 q^{17}-q^{16}+6 q^{15}-5 q^{14}-6 q^{13}+15 q^{12}-5 q^{11}-16 q^{10}+23 q^9-2 q^8-27 q^7+28 q^6+4 q^5-34 q^4+27 q^3+9 q^2-33 q+21+9 q^{-1} -23 q^{-2} +13 q^{-3} +5 q^{-4} -11 q^{-5} +6 q^{-6} + q^{-7} -3 q^{-8} + q^{-9} }
3	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{36}-2 q^{35}-q^{34}+2 q^{33}+5 q^{32}-4 q^{31}-9 q^{30}+3 q^{29}+17 q^{28}-q^{27}-23 q^{26}-7 q^{25}+30 q^{24}+16 q^{23}-34 q^{22}-27 q^{21}+32 q^{20}+41 q^{19}-30 q^{18}-49 q^{17}+21 q^{16}+60 q^{15}-14 q^{14}-65 q^{13}+3 q^{12}+70 q^{11}+8 q^{10}-73 q^9-18 q^8+72 q^7+29 q^6-71 q^5-33 q^4+61 q^3+41 q^2-58 q-34+42 q^{-1} +34 q^{-2} -37 q^{-3} -20 q^{-4} +23 q^{-5} +17 q^{-6} -21 q^{-7} -5 q^{-8} +12 q^{-9} +4 q^{-10} -11 q^{-11} + q^{-12} +6 q^{-13} - q^{-14} -3 q^{-15} - q^{-16} +3 q^{-17} - q^{-18} }
4	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{60}-2 q^{59}-q^{58}+2 q^{57}+q^{56}+6 q^{55}-8 q^{54}-7 q^{53}+2 q^{52}+3 q^{51}+26 q^{50}-12 q^{49}-23 q^{48}-11 q^{47}-5 q^{46}+63 q^{45}+3 q^{44}-29 q^{43}-38 q^{42}-46 q^{41}+93 q^{40}+35 q^{39}-50 q^{37}-112 q^{36}+86 q^{35}+48 q^{34}+58 q^{33}-18 q^{32}-166 q^{31}+49 q^{30}+14 q^{29}+107 q^{28}+52 q^{27}-177 q^{26}+12 q^{25}-58 q^{24}+124 q^{23}+128 q^{22}-152 q^{21}-8 q^{20}-140 q^{19}+115 q^{18}+193 q^{17}-109 q^{16}-20 q^{15}-215 q^{14}+98 q^{13}+243 q^{12}-59 q^{11}-26 q^{10}-273 q^9+67 q^8+266 q^7-4 q^6-15 q^5-295 q^4+22 q^3+240 q^2+34 q+23-256 q^{-1} -20 q^{-2} +164 q^{-3} +35 q^{-4} +61 q^{-5} -170 q^{-6} -30 q^{-7} +80 q^{-8} +3 q^{-9} +69 q^{-10} -82 q^{-11} -14 q^{-12} +28 q^{-13} -23 q^{-14} +48 q^{-15} -29 q^{-16} +2 q^{-17} +8 q^{-18} -25 q^{-19} +23 q^{-20} -8 q^{-21} +5 q^{-22} +3 q^{-23} -12 q^{-24} +6 q^{-25} -2 q^{-26} +3 q^{-27} + q^{-28} -3 q^{-29} + q^{-30} }
5	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{90}-2 q^{89}-q^{88}+2 q^{87}+q^{86}+2 q^{85}+2 q^{84}-6 q^{83}-9 q^{82}+2 q^{81}+7 q^{80}+11 q^{79}+12 q^{78}-9 q^{77}-29 q^{76}-20 q^{75}+6 q^{74}+33 q^{73}+46 q^{72}+15 q^{71}-46 q^{70}-69 q^{69}-41 q^{68}+30 q^{67}+93 q^{66}+84 q^{65}-4 q^{64}-97 q^{63}-122 q^{62}-49 q^{61}+81 q^{60}+149 q^{59}+99 q^{58}-31 q^{57}-145 q^{56}-150 q^{55}-34 q^{54}+112 q^{53}+169 q^{52}+98 q^{51}-36 q^{50}-152 q^{49}-159 q^{48}-51 q^{47}+101 q^{46}+175 q^{45}+145 q^{44}+6 q^{43}-174 q^{42}-234 q^{41}-108 q^{40}+115 q^{39}+290 q^{38}+248 q^{37}-39 q^{36}-334 q^{35}-360 q^{34}-65 q^{33}+337 q^{32}+481 q^{31}+175 q^{30}-335 q^{29}-575 q^{28}-283 q^{27}+314 q^{26}+661 q^{25}+386 q^{24}-294 q^{23}-738 q^{22}-477 q^{21}+273 q^{20}+802 q^{19}+568 q^{18}-251 q^{17}-869 q^{16}-644 q^{15}+225 q^{14}+906 q^{13}+737 q^{12}-183 q^{11}-951 q^{10}-787 q^9+120 q^8+922 q^7+866 q^6-38 q^5-899 q^4-868 q^3-49 q^2+772 q+880+147 q^{-1} -674 q^{-2} -794 q^{-3} -212 q^{-4} +491 q^{-5} +716 q^{-6} +257 q^{-7} -368 q^{-8} -560 q^{-9} -260 q^{-10} +207 q^{-11} +448 q^{-12} +235 q^{-13} -130 q^{-14} -291 q^{-15} -192 q^{-16} +34 q^{-17} +207 q^{-18} +146 q^{-19} -18 q^{-20} -106 q^{-21} -96 q^{-22} -22 q^{-23} +63 q^{-24} +67 q^{-25} +14 q^{-26} -25 q^{-27} -35 q^{-28} -18 q^{-29} +6 q^{-30} +20 q^{-31} +16 q^{-32} -4 q^{-33} -10 q^{-34} -2 q^{-35} -7 q^{-36} +3 q^{-37} +8 q^{-38} -3 q^{-40} +2 q^{-41} -3 q^{-42} - q^{-43} +3 q^{-44} - q^{-45} }
6	Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle q^{126}-2 q^{125}-q^{124}+2 q^{123}+q^{122}+2 q^{121}-2 q^{120}+4 q^{119}-8 q^{118}-9 q^{117}+5 q^{116}+6 q^{115}+12 q^{114}+16 q^{112}-22 q^{111}-35 q^{110}-9 q^{109}+5 q^{108}+36 q^{107}+21 q^{106}+70 q^{105}-21 q^{104}-79 q^{103}-68 q^{102}-48 q^{101}+30 q^{100}+43 q^{99}+193 q^{98}+62 q^{97}-60 q^{96}-133 q^{95}-165 q^{94}-86 q^{93}-49 q^{92}+295 q^{91}+209 q^{90}+108 q^{89}-57 q^{88}-200 q^{87}-245 q^{86}-311 q^{85}+201 q^{84}+214 q^{83}+285 q^{82}+170 q^{81}+27 q^{80}-189 q^{79}-516 q^{78}-37 q^{77}-79 q^{76}+170 q^{75}+243 q^{74}+381 q^{73}+195 q^{72}-341 q^{71}-51 q^{70}-447 q^{69}-301 q^{68}-148 q^{67}+447 q^{66}+615 q^{65}+212 q^{64}+436 q^{63}-453 q^{62}-784 q^{61}-930 q^{60}-15 q^{59}+645 q^{58}+765 q^{57}+1296 q^{56}+92 q^{55}-879 q^{54}-1711 q^{53}-848 q^{52}+151 q^{51}+960 q^{50}+2146 q^{49}+1001 q^{48}-494 q^{47}-2172 q^{46}-1702 q^{45}-668 q^{44}+748 q^{43}+2720 q^{42}+1943 q^{41}+167 q^{40}-2283 q^{39}-2349 q^{38}-1514 q^{37}+326 q^{36}+3007 q^{35}+2713 q^{34}+832 q^{33}-2217 q^{32}-2784 q^{31}-2212 q^{30}-69 q^{29}+3160 q^{28}+3299 q^{27}+1356 q^{26}-2150 q^{25}-3128 q^{24}-2763 q^{23}-352 q^{22}+3297 q^{21}+3795 q^{20}+1786 q^{19}-2085 q^{18}-3443 q^{17}-3262 q^{16}-644 q^{15}+3332 q^{14}+4210 q^{13}+2258 q^{12}-1817 q^{11}-3570 q^{10}-3697 q^9-1116 q^8+3007 q^7+4328 q^6+2736 q^5-1174 q^4-3219 q^3-3810 q^2-1684 q+2192+3857 q^{-1} +2909 q^{-2} -334 q^{-3} -2323 q^{-4} -3321 q^{-5} -1975 q^{-6} +1165 q^{-7} +2814 q^{-8} +2515 q^{-9} +269 q^{-10} -1243 q^{-11} -2327 q^{-12} -1750 q^{-13} +391 q^{-14} +1636 q^{-15} +1707 q^{-16} +415 q^{-17} -438 q^{-18} -1290 q^{-19} -1195 q^{-20} +47 q^{-21} +765 q^{-22} +919 q^{-23} +276 q^{-24} -49 q^{-25} -574 q^{-26} -664 q^{-27} -20 q^{-28} +291 q^{-29} +409 q^{-30} +117 q^{-31} +64 q^{-32} -207 q^{-33} -323 q^{-34} -9 q^{-35} +87 q^{-36} +155 q^{-37} +33 q^{-38} +67 q^{-39} -58 q^{-40} -139 q^{-41} + q^{-42} +14 q^{-43} +50 q^{-44} +2 q^{-45} +41 q^{-46} -11 q^{-47} -49 q^{-48} +4 q^{-49} -3 q^{-50} +14 q^{-51} -5 q^{-52} +17 q^{-53} - q^{-54} -14 q^{-55} +4 q^{-56} -3 q^{-57} +3 q^{-58} -2 q^{-59} +3 q^{-60} + q^{-61} -3 q^{-62} + q^{-63} }

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session, or any of the Computer Talk sections above.

Modifying This Page

Read me first: Modifying Knot Pages See/edit the Rolfsen Knot Page master template (intermediate). See/edit the Rolfsen_Splice_Base (expert). Back to the top.

9_13 9_15