K11a68

K11a67

K11a69

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	K11a68 Quick Notes

K11a68 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X8394 X16,5,17,6 X14,8,15,7 X2,9,3,10 X18,12,19,11 X20,14,21,13 X22,15,1,16 X10,18,11,17 X12,20,13,19 X6,21,7,22
Gauss code	1, -5, 2, -1, 3, -11, 4, -2, 5, -9, 6, -10, 7, -4, 8, -3, 9, -6, 10, -7, 11, -8
Dowker-Thistlethwaite code	4 8 16 14 2 18 20 22 10 12 6
Conway Notation	[.4.21]

Three dimensional invariants

Symmetry type Chiral Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a68/ThurstonBennequinNumber Hyperbolic Volume 14.7003 A-Polynomial See Data:K11a68/A-polynomial

[edit Notes for K11a68's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $4$ Rasmussen s-Invariant -2

[edit Notes for K11a68's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+6t^{3}-14t^{2}+20t-21+20t^{-1}-14t^{-2}+6t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-2z^{6}+2z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 103, 2 }
Jones polynomial	$q^{7}-4q^{6}+7q^{5}-11q^{4}+15q^{3}-16q^{2}+16q-13+10q^{-1}-6q^{-2}+3q^{-3}-q^{-4}$
HOMFLY-PT polynomial (db, data sources)	$-z^{8}a^{-2}-5z^{6}a^{-2}+z^{6}a^{-4}+2z^{6}-a^{2}z^{4}-8z^{4}a^{-2}+3z^{4}a^{-4}+8z^{4}-3a^{2}z^{2}-4z^{2}a^{-2}+z^{2}a^{-4}+8z^{2}-a^{2}+a^{-2}-a^{-4}+2$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-2}+2z^{10}+4az^{9}+10z^{9}a^{-1}+6z^{9}a^{-3}+3a^{2}z^{8}+6z^{8}a^{-2}+8z^{8}a^{-4}+z^{8}+a^{3}z^{7}-14az^{7}-33z^{7}a^{-1}-10z^{7}a^{-3}+8z^{7}a^{-5}-12a^{2}z^{6}-30z^{6}a^{-2}-11z^{6}a^{-4}+7z^{6}a^{-6}-24z^{6}-4a^{3}z^{5}+12az^{5}+32z^{5}a^{-1}+5z^{5}a^{-3}-7z^{5}a^{-5}+4z^{5}a^{-7}+14a^{2}z^{4}+30z^{4}a^{-2}+2z^{4}a^{-4}-7z^{4}a^{-6}+z^{4}a^{-8}+34z^{4}+4a^{3}z^{3}-2az^{3}-10z^{3}a^{-1}-4z^{3}a^{-3}-3z^{3}a^{-5}-3z^{3}a^{-7}-7a^{2}z^{2}-7z^{2}a^{-2}+2z^{2}a^{-4}+z^{2}a^{-6}-15z^{2}-a^{3}z-az+za^{-1}+3za^{-3}+2za^{-5}+a^{2}-a^{-2}-a^{-4}+2$
The A2 invariant	$-q^{12}+q^{8}-q^{6}+2q^{4}-2q^{2}+1+2q^{-2}-q^{-4}+5q^{-6}-2q^{-8}+2q^{-10}-q^{-12}-2q^{-14}+q^{-16}-2q^{-18}+q^{-20}$
The G2 invariant	$q^{60}-2q^{58}+5q^{56}-9q^{54}+11q^{52}-13q^{50}+6q^{48}+11q^{46}-36q^{44}+65q^{42}-85q^{40}+75q^{38}-31q^{36}-52q^{34}+146q^{32}-211q^{30}+217q^{28}-139q^{26}-7q^{24}+171q^{22}-287q^{20}+306q^{18}-209q^{16}+35q^{14}+141q^{12}-251q^{10}+246q^{8}-135q^{6}-24q^{4}+167q^{2}-218+159q^{-2}-20q^{-4}-145q^{-6}+262q^{-8}-285q^{-10}+203q^{-12}-34q^{-14}-159q^{-16}+317q^{-18}-370q^{-20}+309q^{-22}-137q^{-24}-74q^{-26}+250q^{-28}-329q^{-30}+286q^{-32}-138q^{-34}-38q^{-36}+174q^{-38}-207q^{-40}+137q^{-42}-7q^{-44}-119q^{-46}+176q^{-48}-148q^{-50}+46q^{-52}+72q^{-54}-166q^{-56}+201q^{-58}-166q^{-60}+85q^{-62}+10q^{-64}-97q^{-66}+141q^{-68}-155q^{-70}+134q^{-72}-86q^{-74}+29q^{-76}+33q^{-78}-81q^{-80}+104q^{-82}-100q^{-84}+75q^{-86}-37q^{-88}-3q^{-90}+32q^{-92}-49q^{-94}+47q^{-96}-32q^{-98}+18q^{-100}-2q^{-102}-6q^{-104}+9q^{-106}-10q^{-108}+6q^{-110}-3q^{-112}+q^{-114}$

Further Quantum Invariants

K11a68 Quantum Invariants.

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["K11a68"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+6t^{3}-14t^{2}+20t-21+20t^{-1}-14t^{-2}+6t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-2z^{6}+2z^{4}+2z^{2}+1$

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]=

{ 103, 2 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{7}-4q^{6}+7q^{5}-11q^{4}+15q^{3}-16q^{2}+16q-13+10q^{-1}-6q^{-2}+3q^{-3}-q^{-4}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-z^{8}a^{-2}-5z^{6}a^{-2}+z^{6}a^{-4}+2z^{6}-a^{2}z^{4}-8z^{4}a^{-2}+3z^{4}a^{-4}+8z^{4}-3a^{2}z^{2}-4z^{2}a^{-2}+z^{2}a^{-4}+8z^{2}-a^{2}+a^{-2}-a^{-4}+2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-2}+2z^{10}+4az^{9}+10z^{9}a^{-1}+6z^{9}a^{-3}+3a^{2}z^{8}+6z^{8}a^{-2}+8z^{8}a^{-4}+z^{8}+a^{3}z^{7}-14az^{7}-33z^{7}a^{-1}-10z^{7}a^{-3}+8z^{7}a^{-5}-12a^{2}z^{6}-30z^{6}a^{-2}-11z^{6}a^{-4}+7z^{6}a^{-6}-24z^{6}-4a^{3}z^{5}+12az^{5}+32z^{5}a^{-1}+5z^{5}a^{-3}-7z^{5}a^{-5}+4z^{5}a^{-7}+14a^{2}z^{4}+30z^{4}a^{-2}+2z^{4}a^{-4}-7z^{4}a^{-6}+z^{4}a^{-8}+34z^{4}+4a^{3}z^{3}-2az^{3}-10z^{3}a^{-1}-4z^{3}a^{-3}-3z^{3}a^{-5}-3z^{3}a^{-7}-7a^{2}z^{2}-7z^{2}a^{-2}+2z^{2}a^{-4}+z^{2}a^{-6}-15z^{2}-a^{3}z-az+za^{-1}+3za^{-3}+2za^{-5}+a^{2}-a^{-2}-a^{-4}+2$

Vassiliev invariants

V2 and V3:

(2, 1)

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 $8$ $8$ $32$ ${\frac {124}{3}}$ $-{\frac {4}{3}}$ $64$ ${\frac {272}{3}}$ ${\frac {224}{3}}$ $-56$ ${\frac {256}{3}}$ $32$ ${\frac {992}{3}}$ $-{\frac {32}{3}}$ ${\frac {6271}{15}}$ ${\frac {3916}{15}}$ $-{\frac {8156}{45}}$ ${\frac {401}{9}}$ $-{\frac {929}{15}}$

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 $\chi$ (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=$2 is the signature of K11a68. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 6 χ 15                       1 1 13                     3   -3 11                   4 1   3 9                 7 3     -4 7               8 4       4 5             8 7         -1 3           8 8           0 1         6 9             3 -1       4 7               -3 -3     2 6                 4 -5   1 4                   -3 -7   2                     2 -9 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=1$ $i=3$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=-2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=-1$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=0$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{8}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=2$ ${\mathbb {Z} }^{7}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{7}$ ${\mathbb {Z} }^{7}$ $r=4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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[11, Alternating, 68]]
Out[2]=	11
In[3]:=	PD[Knot[11, Alternating, 68]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[16, 5, 17, 6], X[14, 8, 15, 7], X[2, 9, 3, 10], X[18, 12, 19, 11], X[20, 14, 21, 13], X[22, 15, 1, 16], X[10, 18, 11, 17], X[12, 20, 13, 19], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 68]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -11, 4, -2, 5, -9, 6, -10, 7, -4, 8, -3, 9, -6, 10, -7, 11, -8]
In[5]:=	BR[Knot[11, Alternating, 68]]
Out[5]=	BR[Knot[11, Alternating, 68]]
In[6]:=	alex = Alexander[Knot[11, Alternating, 68]][t]
Out[6]=	-4 6 14 20 2 3 4 -21 - t + -- - -- + -- + 20 t - 14 t + 6 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 68]][z]
Out[7]=	2 4 6 8 1 + 2 z + 2 z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 68]}
In[9]:=	{KnotDet[Knot[11, Alternating, 68]], KnotSignature[Knot[11, Alternating, 68]]}
Out[9]=	{103, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 68]][q]
Out[10]=	-4 3 6 10 2 3 4 5 6 -13 - q + -- - -- + -- + 16 q - 16 q + 15 q - 11 q + 7 q - 4 q + 3 2 q q q 7 q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 68], Knot[11, Alternating, 111]}
In[12]:=	A2Invariant[Knot[11, Alternating, 68]][q]
Out[12]=	-12 -8 -6 2 2 2 4 6 8 10 1 - q + q - q + -- - -- + 2 q - q + 5 q - 2 q + 2 q - 4 2 q q 12 14 16 18 20 q - 2 q + q - 2 q + q
In[13]:=	Kauffman[Knot[11, Alternating, 68]][a, z]
Out[13]=	2 2 -4 -2 2 2 z 3 z z 3 2 z 2 z 2 - a - a + a + --- + --- + - - a z - a z - 15 z + -- + ---- - 5 3 a 6 4 a a a a 2 3 3 3 3 7 z 2 2 3 z 3 z 4 z 10 z 3 3 3 ---- - 7 a z - ---- - ---- - ---- - ----- - 2 a z + 4 a z + 2 7 5 3 a a a a a 4 4 4 4 5 5 5 4 z 7 z 2 z 30 z 2 4 4 z 7 z 5 z 34 z + -- - ---- + ---- + ----- + 14 a z + ---- - ---- + ---- + 8 6 4 2 7 5 3 a a a a a a a 5 6 6 6 32 z 5 3 5 6 7 z 11 z 30 z 2 6 ----- + 12 a z - 4 a z - 24 z + ---- - ----- - ----- - 12 a z + a 6 4 2 a a a 7 7 7 8 8 8 z 10 z 33 z 7 3 7 8 8 z 6 z 2 8 ---- - ----- - ----- - 14 a z + a z + z + ---- + ---- + 3 a z + 5 3 a 4 2 a a a a 9 9 10 6 z 10 z 9 10 2 z ---- + ----- + 4 a z + 2 z + ----- 3 a 2 a a
In[14]:=	{Vassiliev[2][Knot[11, Alternating, 68]], Vassiliev[3][Knot[11, Alternating, 68]]}
Out[14]=	{0, 1}
In[15]:=	Kh[Knot[11, Alternating, 68]][q, t]
Out[15]=	3 1 2 1 4 2 6 4 9 q + 8 q + ----- + ----- + ----- + ----- + ----- + ----- + ---- + 9 5 7 4 5 4 5 3 3 3 3 2 2 q t q t q t q t q t q t q t 7 6 q 3 5 5 2 7 2 7 3 9 3 --- + --- + 8 q t + 8 q t + 7 q t + 8 q t + 4 q t + 7 q t + q t t 9 4 11 4 11 5 13 5 15 6 3 q t + 4 q t + q t + 3 q t + q t