9 39

9_38

9_40

(KnotPlot image)	See the full Rolfsen Knot Table. Visit 9 39's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 9 39 at Knotilus!

Knot presentations

Planar diagram presentation	X1627 X3,11,4,10 X7,18,8,1 X17,13,18,12 X9,17,10,16 X5,15,6,14 X15,5,16,4 X11,3,12,2 X13,9,14,8
Gauss code	-1, 8, -2, 7, -6, 1, -3, 9, -5, 2, -8, 4, -9, 6, -7, 5, -4, 3
Dowker-Thistlethwaite code	6 10 14 18 16 2 8 4 12
Conway Notation	[2:2:20]

Minimum Braid Representative	A Morse Link Presentation	An Arc Presentation
Length is 12, width is 5, Braid index is 5		[{11, 6}, {2, 7}, {6, 1}, {8, 3}, {5, 2}, {7, 9}, {4, 8}, {10, 5}, {9, 11}, {3, 10}, {1, 4}]

[edit Notes on presentations of 9 39]

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 39"];

In[4]:=

PD[K]

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

Out[4]=

X1627 X3,11,4,10 X7,18,8,1 X17,13,18,12 X9,17,10,16 X5,15,6,14 X15,5,16,4 X11,3,12,2 X13,9,14,8

In[5]:=

GaussCode[K]

Out[5]=

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

In[6]:=

DTCode[K]

Out[6]=

6 10 14 18 16 2 8 4 12

(The path below may be different on your system)

In[7]:=

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

In[8]:=

ConwayNotation[K]

Out[8]=

[2:2:20]

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,-2,1,4,3,-2,3,4\})}

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, 12, 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, 6}, {2, 7}, {6, 1}, {8, 3}, {5, 2}, {7, 9}, {4, 8}, {10, 5}, {9, 11}, {3, 10}, {1, 4}]

In[14]:=

Draw[ap]

Out[14]=

-Graphics-

Three dimensional invariants

Symmetry type Reversible Unknotting number 1 3-genus 2 Bridge index 3 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 \{4,6\}} Nakanishi index 1 Maximal Thurston-Bennequin number [-1][-10] Hyperbolic Volume 12.8103 A-Polynomial See Data:9 39/A-polynomial

[edit Notes for 9 39'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 2

[edit Notes for 9 39's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-3t^{2}+14t-21+14t^{-1}-3t^{-2}$
Conway polynomial	$-3z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 55, 2 }
Jones polynomial	$-q^{8}+3q^{7}-6q^{6}+8q^{5}-9q^{4}+10q^{3}-8q^{2}+6q-3+q^{-1}$
HOMFLY-PT polynomial (db, data sources)	$-z^{4}a^{-2}-2z^{4}a^{-4}+z^{2}a^{-2}-3z^{2}a^{-4}+3z^{2}a^{-6}+z^{2}+2a^{-2}-2a^{-4}+2a^{-6}-a^{-8}$
Kauffman polynomial (db, data sources)	$2z^{8}a^{-4}+2z^{8}a^{-6}+5z^{7}a^{-3}+9z^{7}a^{-5}+4z^{7}a^{-7}+5z^{6}a^{-2}+5z^{6}a^{-4}+3z^{6}a^{-6}+3z^{6}a^{-8}+3z^{5}a^{-1}-7z^{5}a^{-3}-18z^{5}a^{-5}-7z^{5}a^{-7}+z^{5}a^{-9}-7z^{4}a^{-2}-15z^{4}a^{-4}-13z^{4}a^{-6}-6z^{4}a^{-8}+z^{4}-3z^{3}a^{-1}+5z^{3}a^{-3}+12z^{3}a^{-5}+2z^{3}a^{-7}-2z^{3}a^{-9}+5z^{2}a^{-2}+12z^{2}a^{-4}+9z^{2}a^{-6}+3z^{2}a^{-8}-z^{2}-za^{-3}-3za^{-5}-za^{-7}+za^{-9}-2a^{-2}-2a^{-4}-2a^{-6}-a^{-8}$
The A2 invariant	$q^{4}-q^{2}-1+3q^{-2}-q^{-4}+2q^{-6}+q^{-8}-q^{-10}+q^{-12}-2q^{-14}+2q^{-16}-q^{-20}+2q^{-22}-q^{-24}-q^{-26}$
The G2 invariant	$q^{18}-2q^{16}+4q^{14}-6q^{12}+5q^{10}-3q^{8}-2q^{6}+12q^{4}-19q^{2}+28-30q^{-2}+21q^{-4}-3q^{-6}-27q^{-8}+58q^{-10}-76q^{-12}+73q^{-14}-45q^{-16}-6q^{-18}+63q^{-20}-97q^{-22}+101q^{-24}-61q^{-26}+2q^{-28}+53q^{-30}-80q^{-32}+65q^{-34}-12q^{-36}-45q^{-38}+87q^{-40}-83q^{-42}+36q^{-44}+37q^{-46}-103q^{-48}+134q^{-50}-123q^{-52}+66q^{-54}+10q^{-56}-84q^{-58}+131q^{-60}-134q^{-62}+95q^{-64}-29q^{-66}-43q^{-68}+87q^{-70}-93q^{-72}+59q^{-74}-52q^{-78}+80q^{-80}-61q^{-82}+8q^{-84}+57q^{-86}-100q^{-88}+103q^{-90}-65q^{-92}-q^{-94}+60q^{-96}-93q^{-98}+95q^{-100}-63q^{-102}+19q^{-104}+19q^{-106}-45q^{-108}+45q^{-110}-33q^{-112}+17q^{-114}-3q^{-116}-6q^{-118}+8q^{-120}-8q^{-122}+5q^{-124}-2q^{-126}+q^{-128}$

Further Quantum Invariants

Further quantum knot invariants for 9_39.

A1 Invariants.

Weight	Invariant
1	$q^{3}-2q+3q^{-1}-2q^{-3}+2q^{-5}+q^{-7}-q^{-9}+2q^{-11}-3q^{-13}+2q^{-15}-q^{-17}$
2	$q^{10}-2q^{8}+6q^{4}-8q^{2}-3+18q^{-2}-10q^{-4}-13q^{-6}+21q^{-8}-2q^{-10}-15q^{-12}+11q^{-14}+7q^{-16}-7q^{-18}-6q^{-20}+11q^{-22}+2q^{-24}-18q^{-26}+10q^{-28}+12q^{-30}-20q^{-32}+2q^{-34}+16q^{-36}-11q^{-38}-5q^{-40}+8q^{-42}-q^{-44}-2q^{-46}+q^{-48}$
3	$q^{21}-2q^{19}+3q^{15}-7q^{11}-q^{9}+19q^{7}+4q^{5}-35q^{3}-18q+50q^{-1}+49q^{-3}-58q^{-5}-81q^{-7}+48q^{-9}+110q^{-11}-22q^{-13}-124q^{-15}-8q^{-17}+120q^{-19}+35q^{-21}-92q^{-23}-56q^{-25}+63q^{-27}+66q^{-29}-28q^{-31}-69q^{-33}-7q^{-35}+70q^{-37}+34q^{-39}-68q^{-41}-65q^{-43}+66q^{-45}+87q^{-47}-51q^{-49}-109q^{-51}+28q^{-53}+119q^{-55}+3q^{-57}-116q^{-59}-33q^{-61}+92q^{-63}+58q^{-65}-59q^{-67}-66q^{-69}+28q^{-71}+54q^{-73}-35q^{-77}-11q^{-79}+17q^{-81}+8q^{-83}-5q^{-85}-4q^{-87}+q^{-89}+2q^{-91}-q^{-93}$
4	$q^{36}-2q^{34}+3q^{30}-3q^{28}+q^{26}-5q^{24}+7q^{22}+16q^{20}-16q^{18}-18q^{16}-29q^{14}+34q^{12}+94q^{10}+4q^{8}-88q^{6}-177q^{4}-4q^{2}+269+223q^{-2}-46q^{-4}-458q^{-6}-324q^{-8}+281q^{-10}+587q^{-12}+333q^{-14}-510q^{-16}-757q^{-18}-93q^{-20}+656q^{-22}+779q^{-24}-153q^{-26}-823q^{-28}-512q^{-30}+308q^{-32}+823q^{-34}+256q^{-36}-471q^{-38}-598q^{-40}-94q^{-42}+514q^{-44}+420q^{-46}-67q^{-48}-446q^{-50}-319q^{-52}+170q^{-54}+444q^{-56}+222q^{-58}-308q^{-60}-471q^{-62}-86q^{-64}+492q^{-66}+480q^{-68}-189q^{-70}-631q^{-72}-374q^{-74}+475q^{-76}+743q^{-78}+71q^{-80}-654q^{-82}-709q^{-84}+199q^{-86}+807q^{-88}+460q^{-90}-333q^{-92}-826q^{-94}-250q^{-96}+475q^{-98}+623q^{-100}+160q^{-102}-511q^{-104}-453q^{-106}-18q^{-108}+365q^{-110}+359q^{-112}-71q^{-114}-252q^{-116}-199q^{-118}+31q^{-120}+191q^{-122}+81q^{-124}-19q^{-126}-92q^{-128}-50q^{-130}+31q^{-132}+27q^{-134}+19q^{-136}-11q^{-138}-14q^{-140}+2q^{-142}+q^{-144}+4q^{-146}-q^{-148}-2q^{-150}+q^{-152}$
5	$q^{55}-2q^{53}+3q^{49}-3q^{47}-2q^{45}+3q^{43}+3q^{41}+4q^{39}+3q^{37}-16q^{35}-28q^{33}+q^{31}+45q^{29}+67q^{27}+26q^{25}-78q^{23}-176q^{21}-133q^{19}+107q^{17}+362q^{15}+373q^{13}-4q^{11}-574q^{9}-844q^{7}-376q^{5}+690q^{3}+1486q+1143q^{-1}-420q^{-3}-2090q^{-5}-2322q^{-7}-436q^{-9}+2359q^{-11}+3622q^{-13}+1876q^{-15}-1907q^{-17}-4647q^{-19}-3684q^{-21}+702q^{-23}+4997q^{-25}+5332q^{-27}+1068q^{-29}-4429q^{-31}-6381q^{-33}-2940q^{-35}+3094q^{-37}+6539q^{-39}+4408q^{-41}-1367q^{-43}-5793q^{-45}-5162q^{-47}-316q^{-49}+4443q^{-51}+5172q^{-53}+1581q^{-55}-2899q^{-57}-4539q^{-59}-2348q^{-61}+1461q^{-63}+3655q^{-65}+2656q^{-67}-365q^{-69}-2756q^{-71}-2693q^{-73}-426q^{-75}+2053q^{-77}+2705q^{-79}+986q^{-81}-1633q^{-83}-2831q^{-85}-1472q^{-87}+1364q^{-89}+3173q^{-91}+2078q^{-93}-1183q^{-95}-3652q^{-97}-2847q^{-99}+821q^{-101}+4136q^{-103}+3841q^{-105}-172q^{-107}-4404q^{-109}-4877q^{-111}-889q^{-113}+4209q^{-115}+5774q^{-117}+2249q^{-119}-3374q^{-121}-6210q^{-123}-3713q^{-125}+1949q^{-127}+5923q^{-129}+4882q^{-131}-119q^{-133}-4818q^{-135}-5431q^{-137}-1681q^{-139}+3105q^{-141}+5089q^{-143}+2990q^{-145}-1136q^{-147}-3955q^{-149}-3519q^{-151}-540q^{-153}+2391q^{-155}+3163q^{-157}+1561q^{-159}-845q^{-161}-2237q^{-163}-1830q^{-165}-226q^{-167}+1171q^{-169}+1467q^{-171}+718q^{-173}-315q^{-175}-884q^{-177}-723q^{-179}-124q^{-181}+366q^{-183}+470q^{-185}+238q^{-187}-59q^{-189}-223q^{-191}-183q^{-193}-32q^{-195}+70q^{-197}+84q^{-199}+39q^{-201}-7q^{-203}-33q^{-205}-22q^{-207}+3q^{-209}+8q^{-211}+4q^{-213}+2q^{-215}-q^{-217}-4q^{-219}+q^{-221}+2q^{-223}-q^{-225}$

A2 Invariants.

Weight	Invariant
1,0	$q^{4}-q^{2}-1+3q^{-2}-q^{-4}+2q^{-6}+q^{-8}-q^{-10}+q^{-12}-2q^{-14}+2q^{-16}-q^{-20}+2q^{-22}-q^{-24}-q^{-26}$
1,1	$q^{12}-4q^{10}+10q^{8}-20q^{6}+38q^{4}-62q^{2}+98-150q^{-2}+211q^{-4}-270q^{-6}+334q^{-8}-374q^{-10}+372q^{-12}-316q^{-14}+212q^{-16}-54q^{-18}-136q^{-20}+344q^{-22}-522q^{-24}+662q^{-26}-745q^{-28}+754q^{-30}-702q^{-32}+582q^{-34}-415q^{-36}+218q^{-38}-18q^{-40}-158q^{-42}+298q^{-44}-386q^{-46}+410q^{-48}-380q^{-50}+319q^{-52}-248q^{-54}+168q^{-56}-102q^{-58}+58q^{-60}-28q^{-62}+12q^{-64}-4q^{-66}+q^{-68}$
2,0	$q^{12}-q^{10}-2q^{8}+2q^{6}+5q^{4}-q^{2}-10+14q^{-4}-11q^{-8}+3q^{-10}+11q^{-12}-q^{-14}-10q^{-16}+3q^{-18}+6q^{-20}-3q^{-22}+2q^{-24}+3q^{-26}-2q^{-28}+8q^{-32}-5q^{-34}-10q^{-36}+2q^{-38}+9q^{-40}-4q^{-42}-12q^{-44}+6q^{-46}+9q^{-48}-2q^{-50}-8q^{-52}+6q^{-56}-3q^{-60}-q^{-62}+q^{-64}+q^{-66}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{8}-2q^{6}+5q^{2}-6-q^{-2}+13q^{-4}-10q^{-6}-3q^{-8}+17q^{-10}-10q^{-12}-4q^{-14}+12q^{-16}-4q^{-18}-3q^{-20}+2q^{-22}+5q^{-24}-2q^{-26}-8q^{-28}+8q^{-30}+4q^{-32}-15q^{-34}+9q^{-36}+7q^{-38}-15q^{-40}+7q^{-42}+3q^{-44}-8q^{-46}+4q^{-48}+q^{-50}-2q^{-52}+q^{-54}$
1,0,0	$q^{5}-q^{3}-q^{-1}+3q^{-3}-q^{-5}+3q^{-7}+q^{-9}+q^{-11}-q^{-13}-q^{-15}-2q^{-19}+2q^{-21}+2q^{-25}-q^{-27}+2q^{-29}-q^{-31}-q^{-33}-q^{-35}$

B2 Invariants.

Weight	Invariant
0,1	$q^{8}-2q^{6}+4q^{4}-7q^{2}+10-13q^{-2}+17q^{-4}-16q^{-6}+15q^{-8}-9q^{-10}+4q^{-12}+4q^{-14}-12q^{-16}+20q^{-18}-27q^{-20}+30q^{-22}-31q^{-24}+28q^{-26}-22q^{-28}+16q^{-30}-6q^{-32}-q^{-34}+9q^{-36}-13q^{-38}+15q^{-40}-17q^{-42}+15q^{-44}-12q^{-46}+8q^{-48}-5q^{-50}+2q^{-52}-q^{-54}$
1,0	$q^{14}-2q^{10}-2q^{8}+2q^{6}+6q^{4}+q^{2}-8-7q^{-2}+6q^{-4}+15q^{-6}+q^{-8}-16q^{-10}-9q^{-12}+13q^{-14}+15q^{-16}-5q^{-18}-16q^{-20}+15q^{-24}+5q^{-26}-12q^{-28}-6q^{-30}+10q^{-32}+9q^{-34}-6q^{-36}-10q^{-38}+4q^{-40}+11q^{-42}-2q^{-44}-12q^{-46}-2q^{-48}+13q^{-50}+6q^{-52}-13q^{-54}-14q^{-56}+9q^{-58}+18q^{-60}-q^{-62}-17q^{-64}-8q^{-66}+12q^{-68}+11q^{-70}-5q^{-72}-10q^{-74}-q^{-76}+6q^{-78}+3q^{-80}-2q^{-82}-2q^{-84}+q^{-88}$

G2 Invariants.

Weight

Invariant

1,0

$q^{18}-2q^{16}+4q^{14}-6q^{12}+5q^{10}-3q^{8}-2q^{6}+12q^{4}-19q^{2}+28-30q^{-2}+21q^{-4}-3q^{-6}-27q^{-8}+58q^{-10}-76q^{-12}+73q^{-14}-45q^{-16}-6q^{-18}+63q^{-20}-97q^{-22}+101q^{-24}-61q^{-26}+2q^{-28}+53q^{-30}-80q^{-32}+65q^{-34}-12q^{-36}-45q^{-38}+87q^{-40}-83q^{-42}+36q^{-44}+37q^{-46}-103q^{-48}+134q^{-50}-123q^{-52}+66q^{-54}+10q^{-56}-84q^{-58}+131q^{-60}-134q^{-62}+95q^{-64}-29q^{-66}-43q^{-68}+87q^{-70}-93q^{-72}+59q^{-74}-52q^{-78}+80q^{-80}-61q^{-82}+8q^{-84}+57q^{-86}-100q^{-88}+103q^{-90}-65q^{-92}-q^{-94}+60q^{-96}-93q^{-98}+95q^{-100}-63q^{-102}+19q^{-104}+19q^{-106}-45q^{-108}+45q^{-110}-33q^{-112}+17q^{-114}-3q^{-116}-6q^{-118}+8q^{-120}-8q^{-122}+5q^{-124}-2q^{-126}+q^{-128}$

.

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 39"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-3t^{2}+14t-21+14t^{-1}-3t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

$-3z^{4}+2z^{2}+1$

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]=

{ 55, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{8}+3q^{7}-6q^{6}+8q^{5}-9q^{4}+10q^{3}-8q^{2}+6q-3+q^{-1}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{4}a^{-2}-2z^{4}a^{-4}+z^{2}a^{-2}-3z^{2}a^{-4}+3z^{2}a^{-6}+z^{2}+2a^{-2}-2a^{-4}+2a^{-6}-a^{-8}$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{8}a^{-4}+2z^{8}a^{-6}+5z^{7}a^{-3}+9z^{7}a^{-5}+4z^{7}a^{-7}+5z^{6}a^{-2}+5z^{6}a^{-4}+3z^{6}a^{-6}+3z^{6}a^{-8}+3z^{5}a^{-1}-7z^{5}a^{-3}-18z^{5}a^{-5}-7z^{5}a^{-7}+z^{5}a^{-9}-7z^{4}a^{-2}-15z^{4}a^{-4}-13z^{4}a^{-6}-6z^{4}a^{-8}+z^{4}-3z^{3}a^{-1}+5z^{3}a^{-3}+12z^{3}a^{-5}+2z^{3}a^{-7}-2z^{3}a^{-9}+5z^{2}a^{-2}+12z^{2}a^{-4}+9z^{2}a^{-6}+3z^{2}a^{-8}-z^{2}-za^{-3}-3za^{-5}-za^{-7}+za^{-9}-2a^{-2}-2a^{-4}-2a^{-6}-a^{-8}$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n162,}

Same Jones Polynomial (up to mirroring, $q\leftrightarrow q^{-1}$): {K11n11, K11n112,}

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 39"];

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]=

{ $-3t^{2}+14t-21+14t^{-1}-3t^{-2}$, $-q^{8}+3q^{7}-6q^{6}+8q^{5}-9q^{4}+10q^{3}-8q^{2}+6q-3+q^{-1}$ }

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]=

{K11n162,}

In[6]:=

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

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

Out[6]=

{K11n11, K11n112,}

Vassiliev invariants

V2 and V3:

(2, 4)

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 $8$ $32$ $32$ ${\frac {460}{3}}$ ${\frac {116}{3}}$ $256$ ${\frac {2144}{3}}$ ${\frac {320}{3}}$ $192$ ${\frac {256}{3}}$ $512$ ${\frac {3680}{3}}$ ${\frac {928}{3}}$ ${\frac {49111}{15}}$ $-{\frac {8524}{15}}$ ${\frac {97084}{45}}$ ${\frac {857}{9}}$ ${\frac {5191}{15}}$

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 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 $r$). 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=} 2 is the signature of 9 39. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 6 7 χ 17                   1 -1 15                 2   2 13               4 1   -3 11             4 2     2 9           5 4       -1 7         5 4         1 5       3 5           2 3     3 5             -2 1   1 4               3 -1   2                 -2 -3 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=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}} $r=-1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ 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=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}^{5}\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}^{5}\oplus{\mathbb Z}_2^{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}^{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 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}^{4}\oplus{\mathbb Z}_2^{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}^{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 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}^{4}\oplus{\mathbb Z}_2^{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}^{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 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}^{2}\oplus{\mathbb Z}_2^{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}^{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 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}\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=7} 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^{23}-3 q^{22}+q^{21}+10 q^{20}-16 q^{19}-5 q^{18}+37 q^{17}-30 q^{16}-27 q^{15}+69 q^{14}-32 q^{13}-55 q^{12}+89 q^{11}-23 q^{10}-72 q^9+88 q^8-9 q^7-68 q^6+62 q^5+4 q^4-45 q^3+28 q^2+7 q-17+7 q^{-1} +2 q^{-2} -3 q^{-3} + q^{-4} }
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^{45}+3 q^{44}-q^{43}-5 q^{42}-2 q^{41}+16 q^{40}+8 q^{39}-33 q^{38}-26 q^{37}+51 q^{36}+62 q^{35}-59 q^{34}-120 q^{33}+58 q^{32}+179 q^{31}-25 q^{30}-245 q^{29}-25 q^{28}+298 q^{27}+91 q^{26}-336 q^{25}-162 q^{24}+356 q^{23}+229 q^{22}-357 q^{21}-293 q^{20}+353 q^{19}+331 q^{18}-321 q^{17}-370 q^{16}+291 q^{15}+372 q^{14}-227 q^{13}-373 q^{12}+172 q^{11}+336 q^{10}-100 q^9-288 q^8+44 q^7+220 q^6+2 q^5-156 q^4-18 q^3+91 q^2+25 q-49-17 q^{-1} +23 q^{-2} +8 q^{-3} -10 q^{-4} -2 q^{-5} +3 q^{-6} +2 q^{-7} -3 q^{-8} + q^{-9} }
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^{74}-3 q^{73}+q^{72}+5 q^{71}-3 q^{70}+2 q^{69}-19 q^{68}+4 q^{67}+35 q^{66}+5 q^{65}+6 q^{64}-100 q^{63}-38 q^{62}+108 q^{61}+105 q^{60}+116 q^{59}-260 q^{58}-268 q^{57}+55 q^{56}+286 q^{55}+546 q^{54}-254 q^{53}-651 q^{52}-380 q^{51}+228 q^{50}+1217 q^{49}+209 q^{48}-799 q^{47}-1105 q^{46}-348 q^{45}+1710 q^{44}+1002 q^{43}-452 q^{42}-1713 q^{41}-1256 q^{40}+1765 q^{39}+1727 q^{38}+220 q^{37}-1981 q^{36}-2105 q^{35}+1508 q^{34}+2169 q^{33}+889 q^{32}-1969 q^{31}-2683 q^{30}+1123 q^{29}+2332 q^{28}+1419 q^{27}-1747 q^{26}-2957 q^{25}+634 q^{24}+2205 q^{23}+1798 q^{22}-1260 q^{21}-2863 q^{20}+26 q^{19}+1701 q^{18}+1925 q^{17}-533 q^{16}-2296 q^{15}-489 q^{14}+881 q^{13}+1614 q^{12}+137 q^{11}-1364 q^{10}-612 q^9+132 q^8+950 q^7+384 q^6-521 q^5-358 q^4-174 q^3+345 q^2+250 q-109-89 q^{-1} -128 q^{-2} +72 q^{-3} +77 q^{-4} -20 q^{-5} +3 q^{-6} -38 q^{-7} +12 q^{-8} +14 q^{-9} -9 q^{-10} +5 q^{-11} -6 q^{-12} +3 q^{-13} +2 q^{-14} -3 q^{-15} + q^{-16} }
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^{110}+3 q^{109}-q^{108}-5 q^{107}+3 q^{106}+3 q^{105}+q^{104}+7 q^{103}-6 q^{102}-30 q^{101}-8 q^{100}+29 q^{99}+47 q^{98}+52 q^{97}-20 q^{96}-132 q^{95}-159 q^{94}-11 q^{93}+211 q^{92}+349 q^{91}+212 q^{90}-236 q^{89}-649 q^{88}-610 q^{87}+50 q^{86}+918 q^{85}+1245 q^{84}+513 q^{83}-945 q^{82}-2007 q^{81}-1554 q^{80}+511 q^{79}+2637 q^{78}+2919 q^{77}+657 q^{76}-2779 q^{75}-4485 q^{74}-2468 q^{73}+2201 q^{72}+5738 q^{71}+4783 q^{70}-680 q^{69}-6469 q^{68}-7254 q^{67}-1549 q^{66}+6351 q^{65}+9482 q^{64}+4321 q^{63}-5428 q^{62}-11228 q^{61}-7211 q^{60}+3854 q^{59}+12318 q^{58}+9944 q^{57}-1903 q^{56}-12793 q^{55}-12309 q^{54}-134 q^{53}+12791 q^{52}+14176 q^{51}+2110 q^{50}-12498 q^{49}-15624 q^{48}-3802 q^{47}+11986 q^{46}+16645 q^{45}+5371 q^{44}-11403 q^{43}-17433 q^{42}-6638 q^{41}+10627 q^{40}+17843 q^{39}+7990 q^{38}-9684 q^{37}-18085 q^{36}-9117 q^{35}+8360 q^{34}+17780 q^{33}+10381 q^{32}-6663 q^{31}-17086 q^{30}-11311 q^{29}+4551 q^{28}+15589 q^{27}+12021 q^{26}-2183 q^{25}-13495 q^{24}-12040 q^{23}-208 q^{22}+10743 q^{21}+11390 q^{20}+2243 q^{19}-7720 q^{18}-9909 q^{17}-3653 q^{16}+4709 q^{15}+7949 q^{14}+4195 q^{13}-2223 q^{12}-5645 q^{11}-3988 q^{10}+414 q^9+3563 q^8+3232 q^7+517 q^6-1862 q^5-2242 q^4-849 q^3+768 q^2+1346 q+749-192 q^{-1} -679 q^{-2} -506 q^{-3} -28 q^{-4} +280 q^{-5} +281 q^{-6} +78 q^{-7} -109 q^{-8} -129 q^{-9} -39 q^{-10} +25 q^{-11} +41 q^{-12} +35 q^{-13} -11 q^{-14} -25 q^{-15} +2 q^{-16} +3 q^{-17} -3 q^{-18} +6 q^{-19} + q^{-20} -6 q^{-21} +3 q^{-22} +2 q^{-23} -3 q^{-24} + q^{-25} }

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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