6 3

6_2

7_1

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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]

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]=

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

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

Computer Talk

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

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[6, 3]]
Out[2]=	6
In[3]:=	PD[Knot[6, 3]]
Out[3]=	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[4]:=	GaussCode[Knot[6, 3]]
Out[4]=	GaussCode[1, -6, 2, -1, 4, -5, 6, -2, 3, -4, 5, -3]
In[5]:=	BR[Knot[6, 3]]
Out[5]=	BR[3, {-1, -1, 2, -1, 2, 2}]
In[6]:=	alex = Alexander[Knot[6, 3]][t]
Out[6]=	-2 3 2 5 + t - - - 3 t + t t
In[7]:=	Conway[Knot[6, 3]][z]
Out[7]=	2 4 1 + z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[6, 3], Knot[11, NonAlternating, 12]}
In[9]:=	{KnotDet[Knot[6, 3]], KnotSignature[Knot[6, 3]]}
Out[9]=	{13, 0}
In[10]:=	J=Jones[Knot[6, 3]][q]
Out[10]=	-3 2 2 2 3 3 - q + -- - - - 2 q + 2 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[6, 3]}
In[12]:=	A2Invariant[Knot[6, 3]][q]
Out[12]=	-10 2 2 10 1 - q + -- + 2 q - q 2 q
In[13]:=	Kauffman[Knot[6, 3]][a, z]
Out[13]=	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[14]:=	{Vassiliev[2][Knot[6, 3]], Vassiliev[3][Knot[6, 3]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[6, 3]][q, t]
Out[15]=	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