6 3

6_2

7_1

Visit 6 3's page at the Knot Server (KnotPlot driven, includes 3D interactive images!) Visit 6 3's page at Knotilus! Visit 6 3's page at the original Knot Atlas! 6 3 Quick Notes

3D depiction

Irish knot, sum of four 6.3

Knot presentations

Planar diagram presentation	X4251 X8493 X12,9,1,10 X10,5,11,6 X6,11,7,12 X2837
Gauss code	1, -6, 2, -1, 4, -5, 6, -2, 3, -4, 5, -3
Dowker-Thistlethwaite code	4 8 10 2 12 6
Conway Notation	[2112]

Minimum Braid Representative: Length is 6, width is 3. Braid index is 3.

A Morse Link Presentation:

Three dimensional invariants

Symmetry type Fully amphicheiral Unknotting number 1 3-genus 2 Bridge index 2 Super bridge index $\{3,4\}$ Nakanishi index 1 Maximal Thurston-Bennequin number [-4][-4] Hyperbolic Volume 5.69302 A-Polynomial See Data:6 3/A-polynomial

[edit Notes for 6 3'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 6 3's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{2}-3t+5-3t^{-1}+t^{-2}$
Conway polynomial	$z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 13, 0 }
Jones polynomial	$-q^{3}+2q^{2}-2q+3-2q^{-1}+2q^{-2}-q^{-3}$
HOMFLY-PT polynomial (db, data sources)	$z^{4}-a^{2}z^{2}-z^{2}a^{-2}+3z^{2}-a^{2}-a^{-2}+3$
Kauffman polynomial (db, data sources)	$az^{5}+z^{5}a^{-1}+2a^{2}z^{4}+2z^{4}a^{-2}+4z^{4}+a^{3}z^{3}+az^{3}+z^{3}a^{-1}+z^{3}a^{-3}-3a^{2}z^{2}-3z^{2}a^{-2}-6z^{2}-a^{3}z-2az-2za^{-1}-za^{-3}+a^{2}+a^{-2}+3$
The A2 invariant	$-q^{10}+2q^{2}+1+2q^{-2}-q^{-10}$
The G2 invariant	$q^{52}-q^{50}+2q^{48}-2q^{46}-q^{44}+q^{42}-3q^{40}+4q^{38}-4q^{36}+q^{34}-3q^{30}+3q^{28}-3q^{26}+q^{24}+q^{22}-2q^{20}+q^{18}+q^{16}-q^{14}+4q^{12}-3q^{10}+3q^{8}+q^{6}-q^{4}+6q^{2}-5+6q^{-2}-q^{-4}+q^{-6}+3q^{-8}-3q^{-10}+4q^{-12}-q^{-14}+q^{-16}+q^{-18}-2q^{-20}+q^{-22}+q^{-24}-3q^{-26}+3q^{-28}-3q^{-30}+q^{-34}-4q^{-36}+4q^{-38}-3q^{-40}+q^{-42}-q^{-44}-2q^{-46}+2q^{-48}-q^{-50}+q^{-52}$

Further Quantum Invariants

Further quantum knot invariants for 6_3.

A1 Invariants.

Weight	Invariant
1	$-q^{7}+q^{5}+q+q^{-1}+q^{-5}-q^{-7}$
2	$q^{20}-q^{18}-2q^{16}+2q^{14}-2q^{10}+2q^{8}+q^{6}-q^{4}+q^{2}+1+q^{-2}-q^{-4}+q^{-6}+2q^{-8}-2q^{-10}+2q^{-14}-2q^{-16}-q^{-18}+q^{-20}$
3	$-q^{39}+q^{37}+2q^{35}-3q^{31}-2q^{29}+4q^{27}+2q^{25}-4q^{23}-4q^{21}+3q^{19}+4q^{17}-2q^{15}-4q^{13}+3q^{11}+4q^{9}-2q^{5}+q+q^{-1}-2q^{-5}+4q^{-9}+3q^{-11}-4q^{-13}-2q^{-15}+4q^{-17}+3q^{-19}-4q^{-21}-4q^{-23}+2q^{-25}+4q^{-27}-2q^{-29}-3q^{-31}+2q^{-35}+q^{-37}-q^{-39}$
4	$q^{64}-q^{62}-2q^{60}+q^{56}+5q^{54}-4q^{50}-4q^{48}-2q^{46}+10q^{44}+5q^{42}-4q^{40}-9q^{38}-8q^{36}+9q^{34}+9q^{32}+q^{30}-9q^{28}-11q^{26}+7q^{24}+10q^{22}+4q^{20}-6q^{18}-9q^{16}+3q^{14}+6q^{12}+4q^{10}-2q^{8}-5q^{6}+2q^{2}+3+2q^{-2}-5q^{-6}-2q^{-8}+4q^{-10}+6q^{-12}+3q^{-14}-9q^{-16}-6q^{-18}+4q^{-20}+10q^{-22}+7q^{-24}-11q^{-26}-9q^{-28}+q^{-30}+9q^{-32}+9q^{-34}-8q^{-36}-9q^{-38}-4q^{-40}+5q^{-42}+10q^{-44}-2q^{-46}-4q^{-48}-4q^{-50}+5q^{-54}+q^{-56}-2q^{-60}-q^{-62}+q^{-64}$
5	$-q^{95}+q^{93}+2q^{91}-q^{87}-3q^{85}-3q^{83}+6q^{79}+6q^{77}+q^{75}-6q^{73}-10q^{71}-6q^{69}+5q^{67}+17q^{65}+13q^{63}-3q^{61}-17q^{59}-20q^{57}-5q^{55}+17q^{53}+25q^{51}+9q^{49}-15q^{47}-27q^{45}-17q^{43}+12q^{41}+27q^{39}+19q^{37}-6q^{35}-25q^{33}-19q^{31}+4q^{29}+21q^{27}+17q^{25}-16q^{21}-16q^{19}+11q^{15}+11q^{13}+3q^{11}-6q^{9}-8q^{7}-3q^{5}+3q^{3}+6q+6q^{-1}+3q^{-3}-3q^{-5}-8q^{-7}-6q^{-9}+3q^{-11}+11q^{-13}+11q^{-15}-16q^{-19}-16q^{-21}+17q^{-25}+21q^{-27}+4q^{-29}-19q^{-31}-25q^{-33}-6q^{-35}+19q^{-37}+27q^{-39}+12q^{-41}-17q^{-43}-27q^{-45}-15q^{-47}+9q^{-49}+25q^{-51}+17q^{-53}-5q^{-55}-20q^{-57}-17q^{-59}-3q^{-61}+13q^{-63}+17q^{-65}+5q^{-67}-6q^{-69}-10q^{-71}-6q^{-73}+q^{-75}+6q^{-77}+6q^{-79}-3q^{-83}-3q^{-85}-q^{-87}+2q^{-91}+q^{-93}-q^{-95}$
6	$q^{132}-q^{130}-2q^{128}+q^{124}+3q^{122}+q^{120}+3q^{118}-2q^{116}-8q^{114}-5q^{112}-q^{110}+6q^{108}+8q^{106}+14q^{104}+q^{102}-14q^{100}-19q^{98}-15q^{96}+13q^{92}+38q^{90}+25q^{88}-4q^{86}-29q^{84}-41q^{82}-28q^{80}-q^{78}+49q^{76}+53q^{74}+26q^{72}-18q^{70}-53q^{68}-58q^{66}-30q^{64}+39q^{62}+64q^{60}+53q^{58}+7q^{56}-43q^{54}-66q^{52}-49q^{50}+18q^{48}+53q^{46}+57q^{44}+22q^{42}-24q^{40}-53q^{38}-47q^{36}+4q^{34}+33q^{32}+43q^{30}+21q^{28}-9q^{26}-32q^{24}-33q^{22}-3q^{20}+14q^{18}+23q^{16}+15q^{14}+q^{12}-13q^{10}-17q^{8}-7q^{6}+2q^{4}+11q^{2}+13+11q^{-2}+2q^{-4}-7q^{-6}-17q^{-8}-13q^{-10}+q^{-12}+15q^{-14}+23q^{-16}+14q^{-18}-3q^{-20}-33q^{-22}-32q^{-24}-9q^{-26}+21q^{-28}+43q^{-30}+33q^{-32}+4q^{-34}-47q^{-36}-53q^{-38}-24q^{-40}+22q^{-42}+57q^{-44}+53q^{-46}+18q^{-48}-49q^{-50}-66q^{-52}-43q^{-54}+7q^{-56}+53q^{-58}+64q^{-60}+39q^{-62}-30q^{-64}-58q^{-66}-53q^{-68}-18q^{-70}+26q^{-72}+53q^{-74}+49q^{-76}-q^{-78}-28q^{-80}-41q^{-82}-29q^{-84}-4q^{-86}+25q^{-88}+38q^{-90}+13q^{-92}-15q^{-96}-19q^{-98}-14q^{-100}+q^{-102}+14q^{-104}+8q^{-106}+6q^{-108}-q^{-110}-5q^{-112}-8q^{-114}-2q^{-116}+3q^{-118}+q^{-120}+3q^{-122}+q^{-124}-2q^{-128}-q^{-130}+q^{-132}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{10}+2q^{2}+1+2q^{-2}-q^{-10}$
1,1	$q^{28}-2q^{26}+4q^{24}-6q^{22}+7q^{20}-10q^{18}+8q^{16}-8q^{14}+4q^{12}-4q^{8}+10q^{6}-11q^{4}+18q^{2}-14+18q^{-2}-11q^{-4}+10q^{-6}-4q^{-8}+4q^{-12}-8q^{-14}+8q^{-16}-10q^{-18}+7q^{-20}-6q^{-22}+4q^{-24}-2q^{-26}+q^{-28}$
2,0	$q^{26}-q^{22}-q^{20}-2q^{14}+q^{10}+2q^{4}+2q^{2}+2+2q^{-2}+2q^{-4}+q^{-10}-2q^{-14}-q^{-20}-q^{-22}+q^{-26}$
3,0	$-q^{48}+q^{44}+2q^{42}+q^{40}-2q^{38}-q^{36}+3q^{32}-4q^{28}-4q^{26}-q^{24}+2q^{22}-q^{20}-3q^{18}+4q^{14}+5q^{12}+2q^{10}+2q^{6}+q^{4}-2+q^{-4}+2q^{-6}+2q^{-10}+5q^{-12}+4q^{-14}-3q^{-18}-q^{-20}+2q^{-22}-q^{-24}-4q^{-26}-4q^{-28}+3q^{-32}-q^{-36}-2q^{-38}+q^{-40}+2q^{-42}+q^{-44}-q^{-48}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{22}-q^{20}+q^{16}-3q^{14}-q^{12}-2q^{8}+3q^{4}+3q^{2}+4+3q^{-2}+3q^{-4}-2q^{-8}-q^{-12}-3q^{-14}+q^{-16}-q^{-20}+q^{-22}$
1,0,0	$-q^{13}-q^{9}+2q^{3}+2q+2q^{-1}+2q^{-3}-q^{-9}-q^{-13}$
1,0,1	$q^{36}-2q^{34}+3q^{32}-q^{30}-3q^{28}+7q^{26}-9q^{24}+6q^{22}-q^{20}-9q^{18}+10q^{16}-15q^{14}+5q^{12}-q^{10}-7q^{8}+11q^{6}+8q^{2}+9+8q^{-2}+11q^{-6}-7q^{-8}-q^{-10}+5q^{-12}-15q^{-14}+10q^{-16}-9q^{-18}-q^{-20}+6q^{-22}-9q^{-24}+7q^{-26}-3q^{-28}-q^{-30}+3q^{-32}-2q^{-34}+q^{-36}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{28}+q^{22}-3q^{18}-3q^{16}-2q^{14}-3q^{12}-3q^{10}+4q^{6}+3q^{4}+6q^{2}+8+6q^{-2}+3q^{-4}+4q^{-6}-3q^{-10}-3q^{-12}-2q^{-14}-3q^{-16}-3q^{-18}+q^{-22}+q^{-28}$
1,0,0,0	$-q^{16}-q^{12}-q^{10}+2q^{4}+2q^{2}+3+2q^{-2}+2q^{-4}-q^{-10}-q^{-12}-q^{-16}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{22}+q^{20}-2q^{18}+q^{16}-q^{14}+q^{12}+2q^{6}-q^{4}+3q^{2}-2+3q^{-2}-q^{-4}+2q^{-6}+q^{-12}-q^{-14}+q^{-16}-2q^{-18}+q^{-20}-q^{-22}$
1,0	$q^{36}-q^{32}-q^{30}+q^{28}+q^{26}-q^{24}-2q^{22}-q^{20}+q^{18}-q^{14}-q^{12}+q^{10}+q^{8}+q^{6}+2q^{2}+3+2q^{-2}+q^{-6}+q^{-8}+q^{-10}-q^{-12}-q^{-14}+q^{-18}-q^{-20}-2q^{-22}-q^{-24}+q^{-26}+q^{-28}-q^{-30}-q^{-32}+q^{-36}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{30}-q^{28}+q^{26}-q^{24}+q^{22}-2q^{20}-q^{18}-2q^{16}-q^{14}-q^{12}-2q^{10}+q^{8}+q^{6}+5q^{4}+2q^{2}+6+2q^{-2}+5q^{-4}+q^{-6}+q^{-8}-2q^{-10}-q^{-12}-q^{-14}-2q^{-16}-q^{-18}-2q^{-20}+q^{-22}-q^{-24}+q^{-26}-q^{-28}+q^{-30}$

G2 Invariants.

Weight	Invariant
1,0	$q^{52}-q^{50}+2q^{48}-2q^{46}-q^{44}+q^{42}-3q^{40}+4q^{38}-4q^{36}+q^{34}-3q^{30}+3q^{28}-3q^{26}+q^{24}+q^{22}-2q^{20}+q^{18}+q^{16}-q^{14}+4q^{12}-3q^{10}+3q^{8}+q^{6}-q^{4}+6q^{2}-5+6q^{-2}-q^{-4}+q^{-6}+3q^{-8}-3q^{-10}+4q^{-12}-q^{-14}+q^{-16}+q^{-18}-2q^{-20}+q^{-22}+q^{-24}-3q^{-26}+3q^{-28}-3q^{-30}+q^{-34}-4q^{-36}+4q^{-38}-3q^{-40}+q^{-42}-q^{-44}-2q^{-46}+2q^{-48}-q^{-50}+q^{-52}$

.

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["6 3"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{2}-3t+5-3t^{-1}+t^{-2}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 13, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{3}+2q^{2}-2q+3-2q^{-1}+2q^{-2}-q^{-3}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$az^{5}+z^{5}a^{-1}+2a^{2}z^{4}+2z^{4}a^{-2}+4z^{4}+a^{3}z^{3}+az^{3}+z^{3}a^{-1}+z^{3}a^{-3}-3a^{2}z^{2}-3z^{2}a^{-2}-6z^{2}-a^{3}z-2az-2za^{-1}-za^{-3}+a^{2}+a^{-2}+3$

"Similar" Knots (within the Atlas)

Same Alexander/Conway Polynomial: {K11n12, ...}

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

Vassiliev invariants

V2 and V3:

(1, 0)

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 $4$ $0$ $8$ ${\frac {14}{3}}$ $-{\frac {14}{3}}$ $0$ $0$ $0$ $0$ ${\frac {32}{3}}$ $0$ ${\frac {56}{3}}$ $-{\frac {56}{3}}$ ${\frac {511}{30}}$ ${\frac {418}{15}}$ $-{\frac {1858}{45}}$ ${\frac {65}{18}}$ $-{\frac {449}{30}}$

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 6 3. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -3 -2 -1 0 1 2 3 χ 7             1 -1 5           1   1 3         1 1   0 1       2 1     1 -1     1 2       1 -3   1 1         0 -5   1           1 -7 1             -1

Integral Khovanov Homology (db, data source)

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

The Coloured Jones Polynomials

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

$n$	$J_{n}$
2	$q^{9}-2q^{8}-q^{7}+5q^{6}-4q^{5}-3q^{4}+9q^{3}-5q^{2}-5q+11-5q^{-1}-5q^{-2}+9q^{-3}-3q^{-4}-4q^{-5}+5q^{-6}-q^{-7}-2q^{-8}+q^{-9}$
3	$-q^{18}+2q^{17}+q^{16}-2q^{15}-4q^{14}+3q^{13}+7q^{12}-4q^{11}-10q^{10}+3q^{9}+14q^{8}-3q^{7}-16q^{6}+q^{5}+21q^{4}-2q^{3}-20q^{2}-q+23-q^{-1}-20q^{-2}-2q^{-3}+21q^{-4}+q^{-5}-16q^{-6}-3q^{-7}+14q^{-8}+3q^{-9}-10q^{-10}-4q^{-11}+7q^{-12}+3q^{-13}-4q^{-14}-2q^{-15}+q^{-16}+2q^{-17}-q^{-18}$
4	$q^{30}-2q^{29}-q^{28}+2q^{27}+q^{26}+5q^{25}-7q^{24}-5q^{23}+2q^{22}+3q^{21}+17q^{20}-12q^{19}-14q^{18}-3q^{17}+4q^{16}+34q^{15}-12q^{14}-22q^{13}-13q^{12}+2q^{11}+52q^{10}-9q^{9}-28q^{8}-23q^{7}-q^{6}+64q^{5}-6q^{4}-30q^{3}-29q^{2}-4q+69-4q^{-1}-29q^{-2}-30q^{-3}-6q^{-4}+64q^{-5}-q^{-6}-23q^{-7}-28q^{-8}-9q^{-9}+52q^{-10}+2q^{-11}-13q^{-12}-22q^{-13}-12q^{-14}+34q^{-15}+4q^{-16}-3q^{-17}-14q^{-18}-12q^{-19}+17q^{-20}+3q^{-21}+2q^{-22}-5q^{-23}-7q^{-24}+5q^{-25}+q^{-26}+2q^{-27}-q^{-28}-2q^{-29}+q^{-30}$
5	$-q^{45}+2q^{44}+q^{43}-2q^{42}-q^{41}-2q^{40}-q^{39}+5q^{38}+7q^{37}-2q^{36}-6q^{35}-9q^{34}-5q^{33}+9q^{32}+18q^{31}+10q^{30}-10q^{29}-25q^{28}-19q^{27}+6q^{26}+33q^{25}+32q^{24}-2q^{23}-41q^{22}-43q^{21}-6q^{20}+43q^{19}+61q^{18}+13q^{17}-49q^{16}-68q^{15}-25q^{14}+49q^{13}+84q^{12}+30q^{11}-53q^{10}-85q^{9}-41q^{8}+49q^{7}+100q^{6}+41q^{5}-53q^{4}-93q^{3}-50q^{2}+47q+105+47q^{-1}-50q^{-2}-93q^{-3}-53q^{-4}+41q^{-5}+100q^{-6}+49q^{-7}-41q^{-8}-85q^{-9}-53q^{-10}+30q^{-11}+84q^{-12}+49q^{-13}-25q^{-14}-68q^{-15}-49q^{-16}+13q^{-17}+61q^{-18}+43q^{-19}-6q^{-20}-43q^{-21}-41q^{-22}-2q^{-23}+32q^{-24}+33q^{-25}+6q^{-26}-19q^{-27}-25q^{-28}-10q^{-29}+10q^{-30}+18q^{-31}+9q^{-32}-5q^{-33}-9q^{-34}-6q^{-35}-2q^{-36}+7q^{-37}+5q^{-38}-q^{-39}-2q^{-40}-q^{-41}-2q^{-42}+q^{-43}+2q^{-44}-q^{-45}$
6	$q^{63}-2q^{62}-q^{61}+2q^{60}+q^{59}+2q^{58}-2q^{57}+3q^{56}-7q^{55}-7q^{54}+5q^{53}+5q^{52}+9q^{51}+9q^{49}-20q^{48}-22q^{47}+9q^{45}+24q^{44}+13q^{43}+34q^{42}-33q^{41}-51q^{40}-25q^{39}-3q^{38}+37q^{37}+40q^{36}+84q^{35}-29q^{34}-78q^{33}-69q^{32}-38q^{31}+32q^{30}+68q^{29}+153q^{28}-4q^{27}-89q^{26}-115q^{25}-88q^{24}+9q^{23}+85q^{22}+220q^{21}+31q^{20}-85q^{19}-150q^{18}-134q^{17}-20q^{16}+91q^{15}+271q^{14}+60q^{13}-75q^{12}-172q^{11}-164q^{10}-43q^{9}+90q^{8}+301q^{7}+77q^{6}-66q^{5}-180q^{4}-178q^{3}-57q^{2}+86q+311+86q^{-1}-57q^{-2}-178q^{-3}-180q^{-4}-66q^{-5}+77q^{-6}+301q^{-7}+90q^{-8}-43q^{-9}-164q^{-10}-172q^{-11}-75q^{-12}+60q^{-13}+271q^{-14}+91q^{-15}-20q^{-16}-134q^{-17}-150q^{-18}-85q^{-19}+31q^{-20}+220q^{-21}+85q^{-22}+9q^{-23}-88q^{-24}-115q^{-25}-89q^{-26}-4q^{-27}+153q^{-28}+68q^{-29}+32q^{-30}-38q^{-31}-69q^{-32}-78q^{-33}-29q^{-34}+84q^{-35}+40q^{-36}+37q^{-37}-3q^{-38}-25q^{-39}-51q^{-40}-33q^{-41}+34q^{-42}+13q^{-43}+24q^{-44}+9q^{-45}-22q^{-47}-20q^{-48}+9q^{-49}+9q^{-51}+5q^{-52}+5q^{-53}-7q^{-54}-7q^{-55}+3q^{-56}-2q^{-57}+2q^{-58}+q^{-59}+2q^{-60}-q^{-61}-2q^{-62}+q^{-63}$
7	$-q^{84}+2q^{83}+q^{82}-2q^{81}-q^{80}-2q^{79}+2q^{78}-q^{76}+7q^{75}+4q^{74}-4q^{73}-5q^{72}-11q^{71}-q^{70}+3q^{69}-q^{68}+21q^{67}+16q^{66}+q^{65}-9q^{64}-34q^{63}-21q^{62}-8q^{61}-4q^{60}+44q^{59}+49q^{58}+30q^{57}+12q^{56}-59q^{55}-68q^{54}-55q^{53}-41q^{52}+58q^{51}+94q^{50}+98q^{49}+78q^{48}-50q^{47}-118q^{46}-138q^{45}-128q^{44}+28q^{43}+126q^{42}+181q^{41}+195q^{40}+7q^{39}-135q^{38}-224q^{37}-250q^{36}-50q^{35}+119q^{34}+256q^{33}+322q^{32}+101q^{31}-115q^{30}-281q^{29}-371q^{28}-149q^{27}+86q^{26}+299q^{25}+433q^{24}+191q^{23}-75q^{22}-312q^{21}-460q^{20}-232q^{19}+45q^{18}+319q^{17}+506q^{16}+260q^{15}-44q^{14}-321q^{13}-512q^{12}-284q^{11}+14q^{10}+322q^{9}+547q^{8}+298q^{7}-23q^{6}-320q^{5}-534q^{4}-309q^{3}-7q^{2}+316q+559+316q^{-1}-7q^{-2}-309q^{-3}-534q^{-4}-320q^{-5}-23q^{-6}+298q^{-7}+547q^{-8}+322q^{-9}+14q^{-10}-284q^{-11}-512q^{-12}-321q^{-13}-44q^{-14}+260q^{-15}+506q^{-16}+319q^{-17}+45q^{-18}-232q^{-19}-460q^{-20}-312q^{-21}-75q^{-22}+191q^{-23}+433q^{-24}+299q^{-25}+86q^{-26}-149q^{-27}-371q^{-28}-281q^{-29}-115q^{-30}+101q^{-31}+322q^{-32}+256q^{-33}+119q^{-34}-50q^{-35}-250q^{-36}-224q^{-37}-135q^{-38}+7q^{-39}+195q^{-40}+181q^{-41}+126q^{-42}+28q^{-43}-128q^{-44}-138q^{-45}-118q^{-46}-50q^{-47}+78q^{-48}+98q^{-49}+94q^{-50}+58q^{-51}-41q^{-52}-55q^{-53}-68q^{-54}-59q^{-55}+12q^{-56}+30q^{-57}+49q^{-58}+44q^{-59}-4q^{-60}-8q^{-61}-21q^{-62}-34q^{-63}-9q^{-64}+q^{-65}+16q^{-66}+21q^{-67}-q^{-68}+3q^{-69}-q^{-70}-11q^{-71}-5q^{-72}-4q^{-73}+4q^{-74}+7q^{-75}-q^{-76}+2q^{-78}-2q^{-79}-q^{-80}-2q^{-81}+q^{-82}+2q^{-83}-q^{-84}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 29, 2005, 15:27:48)...
In[2]:=	PD[Knot[6, 3]]
Out[2]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[12, 9, 1, 10], X[10, 5, 11, 6], X[6, 11, 7, 12], X[2, 8, 3, 7]]
In[3]:=	GaussCode[Knot[6, 3]]
Out[3]=	GaussCode[1, -6, 2, -1, 4, -5, 6, -2, 3, -4, 5, -3]
In[4]:=	DTCode[Knot[6, 3]]
Out[4]=	DTCode[4, 8, 10, 2, 12, 6]
In[5]:=	br = BR[Knot[6, 3]]
Out[5]=	BR[3, {-1, -1, 2, -1, 2, 2}]
In[6]:=	{First[br], Crossings[br]}
Out[6]=	{3, 6}
In[7]:=	BraidIndex[Knot[6, 3]]
Out[7]=	3
In[8]:=	Show[DrawMorseLink[Knot[6, 3]]]
Out[8]=	-Graphics-
In[9]:=	(#[Knot[6, 3]]&) /@ {SymmetryType, UnknottingNumber, ThreeGenus, BridgeIndex, SuperBridgeIndex, NakanishiIndex}
Out[9]=	{FullyAmphicheiral, 1, 2, 2, {3, 4}, 1}
In[10]:=	alex = Alexander[Knot[6, 3]][t]
Out[10]=	-2 3 2 5 + t - - - 3 t + t t
In[11]:=	Conway[Knot[6, 3]][z]
Out[11]=	2 4 1 + z + z
In[12]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[12]=	{Knot[6, 3], Knot[11, NonAlternating, 12]}
In[13]:=	{KnotDet[Knot[6, 3]], KnotSignature[Knot[6, 3]]}
Out[13]=	{13, 0}
In[14]:=	Jones[Knot[6, 3]][q]
Out[14]=	-3 2 2 2 3 3 - q + -- - - - 2 q + 2 q - q 2 q q
In[15]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[15]=	{Knot[6, 3]}
In[16]:=	A2Invariant[Knot[6, 3]][q]
Out[16]=	-10 2 2 10 1 - q + -- + 2 q - q 2 q
In[17]:=	HOMFLYPT[Knot[6, 3]][a, z]
Out[17]=	2 -2 2 2 z 2 2 4 3 - a - a + 3 z - -- - a z + z 2 a
In[18]:=	Kauffman[Knot[6, 3]][a, z]
Out[18]=	2 3 -2 2 z 2 z 3 2 3 z 2 2 z 3 + a + a - -- - --- - 2 a z - a z - 6 z - ---- - 3 a z + -- + 3 a 2 3 a a a 3 4 5 z 3 3 3 4 2 z 2 4 z 5 -- + a z + a z + 4 z + ---- + 2 a z + -- + a z a 2 a a
In[19]:=	{Vassiliev[2][Knot[6, 3]], Vassiliev[3][Knot[6, 3]]}
Out[19]=	{1, 0}
In[20]:=	Kh[Knot[6, 3]][q, t]
Out[20]=	2 1 1 1 1 1 3 3 2 - + 2 q + ----- + ----- + ----- + ---- + --- + q t + q t + q t + q 7 3 5 2 3 2 3 q t q t q t q t q t 5 2 7 3 q t + q t
In[21]:=	ColouredJones[Knot[6, 3], 2][q]
Out[21]=	-9 2 -7 5 4 3 9 5 5 2 3 11 + q - -- - q + -- - -- - -- + -- - -- - - - 5 q - 5 q + 9 q - 8 6 5 4 3 2 q q q q q q q 4 5 6 7 8 9 3 q - 4 q + 5 q - q - 2 q + q

See/edit the Rolfsen_Splice_Template. Back to the top.

6_2 7_1