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]=

${\textrm {BR}}(5,\{1,1,2,-1,-3,2,-3,4,-3,4\})$

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 $\{4,7\}$ 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 $1$ Topological 4 genus $1$ Concordance genus $2$ Rasmussen s-Invariant 0

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

Polynomial invariants

Alexander polynomial	$2t^{2}-9t+15-9t^{-1}+2t^{-2}$
Conway polynomial	$2z^{4}-z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
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)	$z^{4}a^{-2}+z^{4}-a^{2}z^{2}+z^{2}a^{-2}-2z^{2}a^{-4}+z^{2}+a^{-2}-2a^{-4}+a^{-6}+1$
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]=

$2t^{2}-9t+15-9t^{-1}+2t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$2z^{4}-z^{2}+1$

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]=

$\{1\}$

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]=

$z^{4}a^{-2}+z^{4}-a^{2}z^{2}+z^{2}a^{-2}-2z^{2}a^{-4}+z^{2}+a^{-2}-2a^{-4}+a^{-6}+1$

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 $-4$ $-16$ $8$ $-{\frac {110}{3}}$ $-{\frac {34}{3}}$ $64$ ${\frac {32}{3}}$ ${\frac {128}{3}}$ $-48$ $-{\frac {32}{3}}$ $128$ ${\frac {440}{3}}$ ${\frac {136}{3}}$ ${\frac {10529}{30}}$ ${\frac {1502}{15}}$ ${\frac {4978}{45}}$ $-{\frac {833}{18}}$ ${\frac {449}{30}}$

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 $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). The squares with yellow highlighting are those on the "critical diagonals", where $j-2r=s+1$ or $j-2r=s-1$, where $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)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-3$ ${\mathbb {Z} }$ $r=-2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=0$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{3}$ $r=1$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=2$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=3$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

The Coloured Jones Polynomials

The Coloured Jones Polynomials (in the $n+1$-dimensional representation of $sl(2)$)

$n$	$J_{n}$
2	$q^{18}-2q^{17}-q^{16}+6q^{15}-5q^{14}-6q^{13}+15q^{12}-5q^{11}-16q^{10}+23q^{9}-2q^{8}-27q^{7}+28q^{6}+4q^{5}-34q^{4}+27q^{3}+9q^{2}-33q+21+9q^{-1}-23q^{-2}+13q^{-3}+5q^{-4}-11q^{-5}+6q^{-6}+q^{-7}-3q^{-8}+q^{-9}$
3	$q^{36}-2q^{35}-q^{34}+2q^{33}+5q^{32}-4q^{31}-9q^{30}+3q^{29}+17q^{28}-q^{27}-23q^{26}-7q^{25}+30q^{24}+16q^{23}-34q^{22}-27q^{21}+32q^{20}+41q^{19}-30q^{18}-49q^{17}+21q^{16}+60q^{15}-14q^{14}-65q^{13}+3q^{12}+70q^{11}+8q^{10}-73q^{9}-18q^{8}+72q^{7}+29q^{6}-71q^{5}-33q^{4}+61q^{3}+41q^{2}-58q-34+42q^{-1}+34q^{-2}-37q^{-3}-20q^{-4}+23q^{-5}+17q^{-6}-21q^{-7}-5q^{-8}+12q^{-9}+4q^{-10}-11q^{-11}+q^{-12}+6q^{-13}-q^{-14}-3q^{-15}-q^{-16}+3q^{-17}-q^{-18}$
4	$q^{60}-2q^{59}-q^{58}+2q^{57}+q^{56}+6q^{55}-8q^{54}-7q^{53}+2q^{52}+3q^{51}+26q^{50}-12q^{49}-23q^{48}-11q^{47}-5q^{46}+63q^{45}+3q^{44}-29q^{43}-38q^{42}-46q^{41}+93q^{40}+35q^{39}-50q^{37}-112q^{36}+86q^{35}+48q^{34}+58q^{33}-18q^{32}-166q^{31}+49q^{30}+14q^{29}+107q^{28}+52q^{27}-177q^{26}+12q^{25}-58q^{24}+124q^{23}+128q^{22}-152q^{21}-8q^{20}-140q^{19}+115q^{18}+193q^{17}-109q^{16}-20q^{15}-215q^{14}+98q^{13}+243q^{12}-59q^{11}-26q^{10}-273q^{9}+67q^{8}+266q^{7}-4q^{6}-15q^{5}-295q^{4}+22q^{3}+240q^{2}+34q+23-256q^{-1}-20q^{-2}+164q^{-3}+35q^{-4}+61q^{-5}-170q^{-6}-30q^{-7}+80q^{-8}+3q^{-9}+69q^{-10}-82q^{-11}-14q^{-12}+28q^{-13}-23q^{-14}+48q^{-15}-29q^{-16}+2q^{-17}+8q^{-18}-25q^{-19}+23q^{-20}-8q^{-21}+5q^{-22}+3q^{-23}-12q^{-24}+6q^{-25}-2q^{-26}+3q^{-27}+q^{-28}-3q^{-29}+q^{-30}$
5	$q^{90}-2q^{89}-q^{88}+2q^{87}+q^{86}+2q^{85}+2q^{84}-6q^{83}-9q^{82}+2q^{81}+7q^{80}+11q^{79}+12q^{78}-9q^{77}-29q^{76}-20q^{75}+6q^{74}+33q^{73}+46q^{72}+15q^{71}-46q^{70}-69q^{69}-41q^{68}+30q^{67}+93q^{66}+84q^{65}-4q^{64}-97q^{63}-122q^{62}-49q^{61}+81q^{60}+149q^{59}+99q^{58}-31q^{57}-145q^{56}-150q^{55}-34q^{54}+112q^{53}+169q^{52}+98q^{51}-36q^{50}-152q^{49}-159q^{48}-51q^{47}+101q^{46}+175q^{45}+145q^{44}+6q^{43}-174q^{42}-234q^{41}-108q^{40}+115q^{39}+290q^{38}+248q^{37}-39q^{36}-334q^{35}-360q^{34}-65q^{33}+337q^{32}+481q^{31}+175q^{30}-335q^{29}-575q^{28}-283q^{27}+314q^{26}+661q^{25}+386q^{24}-294q^{23}-738q^{22}-477q^{21}+273q^{20}+802q^{19}+568q^{18}-251q^{17}-869q^{16}-644q^{15}+225q^{14}+906q^{13}+737q^{12}-183q^{11}-951q^{10}-787q^{9}+120q^{8}+922q^{7}+866q^{6}-38q^{5}-899q^{4}-868q^{3}-49q^{2}+772q+880+147q^{-1}-674q^{-2}-794q^{-3}-212q^{-4}+491q^{-5}+716q^{-6}+257q^{-7}-368q^{-8}-560q^{-9}-260q^{-10}+207q^{-11}+448q^{-12}+235q^{-13}-130q^{-14}-291q^{-15}-192q^{-16}+34q^{-17}+207q^{-18}+146q^{-19}-18q^{-20}-106q^{-21}-96q^{-22}-22q^{-23}+63q^{-24}+67q^{-25}+14q^{-26}-25q^{-27}-35q^{-28}-18q^{-29}+6q^{-30}+20q^{-31}+16q^{-32}-4q^{-33}-10q^{-34}-2q^{-35}-7q^{-36}+3q^{-37}+8q^{-38}-3q^{-40}+2q^{-41}-3q^{-42}-q^{-43}+3q^{-44}-q^{-45}$
6	$q^{126}-2q^{125}-q^{124}+2q^{123}+q^{122}+2q^{121}-2q^{120}+4q^{119}-8q^{118}-9q^{117}+5q^{116}+6q^{115}+12q^{114}+16q^{112}-22q^{111}-35q^{110}-9q^{109}+5q^{108}+36q^{107}+21q^{106}+70q^{105}-21q^{104}-79q^{103}-68q^{102}-48q^{101}+30q^{100}+43q^{99}+193q^{98}+62q^{97}-60q^{96}-133q^{95}-165q^{94}-86q^{93}-49q^{92}+295q^{91}+209q^{90}+108q^{89}-57q^{88}-200q^{87}-245q^{86}-311q^{85}+201q^{84}+214q^{83}+285q^{82}+170q^{81}+27q^{80}-189q^{79}-516q^{78}-37q^{77}-79q^{76}+170q^{75}+243q^{74}+381q^{73}+195q^{72}-341q^{71}-51q^{70}-447q^{69}-301q^{68}-148q^{67}+447q^{66}+615q^{65}+212q^{64}+436q^{63}-453q^{62}-784q^{61}-930q^{60}-15q^{59}+645q^{58}+765q^{57}+1296q^{56}+92q^{55}-879q^{54}-1711q^{53}-848q^{52}+151q^{51}+960q^{50}+2146q^{49}+1001q^{48}-494q^{47}-2172q^{46}-1702q^{45}-668q^{44}+748q^{43}+2720q^{42}+1943q^{41}+167q^{40}-2283q^{39}-2349q^{38}-1514q^{37}+326q^{36}+3007q^{35}+2713q^{34}+832q^{33}-2217q^{32}-2784q^{31}-2212q^{30}-69q^{29}+3160q^{28}+3299q^{27}+1356q^{26}-2150q^{25}-3128q^{24}-2763q^{23}-352q^{22}+3297q^{21}+3795q^{20}+1786q^{19}-2085q^{18}-3443q^{17}-3262q^{16}-644q^{15}+3332q^{14}+4210q^{13}+2258q^{12}-1817q^{11}-3570q^{10}-3697q^{9}-1116q^{8}+3007q^{7}+4328q^{6}+2736q^{5}-1174q^{4}-3219q^{3}-3810q^{2}-1684q+2192+3857q^{-1}+2909q^{-2}-334q^{-3}-2323q^{-4}-3321q^{-5}-1975q^{-6}+1165q^{-7}+2814q^{-8}+2515q^{-9}+269q^{-10}-1243q^{-11}-2327q^{-12}-1750q^{-13}+391q^{-14}+1636q^{-15}+1707q^{-16}+415q^{-17}-438q^{-18}-1290q^{-19}-1195q^{-20}+47q^{-21}+765q^{-22}+919q^{-23}+276q^{-24}-49q^{-25}-574q^{-26}-664q^{-27}-20q^{-28}+291q^{-29}+409q^{-30}+117q^{-31}+64q^{-32}-207q^{-33}-323q^{-34}-9q^{-35}+87q^{-36}+155q^{-37}+33q^{-38}+67q^{-39}-58q^{-40}-139q^{-41}+q^{-42}+14q^{-43}+50q^{-44}+2q^{-45}+41q^{-46}-11q^{-47}-49q^{-48}+4q^{-49}-3q^{-50}+14q^{-51}-5q^{-52}+17q^{-53}-q^{-54}-14q^{-55}+4q^{-56}-3q^{-57}+3q^{-58}-2q^{-59}+3q^{-60}+q^{-61}-3q^{-62}+q^{-63}$

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.

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