K11a72

K11a71

K11a73

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	K11a72 Quick Notes

K11a72 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,4,11,3 X12,5,13,6 X14,8,15,7 X2,10,3,9 X22,11,1,12 X18,14,19,13 X20,15,21,16 X8,18,9,17 X6,19,7,20 X16,21,17,22
Gauss code	1, -5, 2, -1, 3, -10, 4, -9, 5, -2, 6, -3, 7, -4, 8, -11, 9, -7, 10, -8, 11, -6
Dowker-Thistlethwaite code	4 10 12 14 2 22 18 20 8 6 16
Conway Notation	[.2111.20]

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:K11a72/ThurstonBennequinNumber Hyperbolic Volume 17.173 A-Polynomial See Data:K11a72/A-polynomial

[edit Notes for K11a72's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus Missing Topological 4 genus Missing Concordance genus $[2,4]$ Rasmussen s-Invariant 0

[edit Notes for K11a72's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$t^{4}-6t^{3}+18t^{2}-32t+39-32t^{-1}+18t^{-2}-6t^{-3}+t^{-4}$
Conway polynomial	$z^{8}+2z^{6}+2z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 153, 0 }
Jones polynomial	$q^{6}-5q^{5}+10q^{4}-16q^{3}+22q^{2}-24q+25-21q^{-1}+15q^{-2}-9q^{-3}+4q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+5z^{6}-3a^{2}z^{4}-6z^{4}a^{-2}+z^{4}a^{-4}+10z^{4}-3a^{2}z^{2}-4z^{2}a^{-2}+z^{2}a^{-4}+8z^{2}-a^{2}+a^{-2}-a^{-4}+2$
Kauffman polynomial (db, data sources)	$2z^{10}a^{-2}+2z^{10}+7az^{9}+14z^{9}a^{-1}+7z^{9}a^{-3}+10a^{2}z^{8}+19z^{8}a^{-2}+9z^{8}a^{-4}+20z^{8}+8a^{3}z^{7}+az^{7}-16z^{7}a^{-1}-4z^{7}a^{-3}+5z^{7}a^{-5}+4a^{4}z^{6}-15a^{2}z^{6}-55z^{6}a^{-2}-20z^{6}a^{-4}+z^{6}a^{-6}-53z^{6}+a^{5}z^{5}-12a^{3}z^{5}-20az^{5}-15z^{5}a^{-1}-18z^{5}a^{-3}-10z^{5}a^{-5}-5a^{4}z^{4}+11a^{2}z^{4}+44z^{4}a^{-2}+12z^{4}a^{-4}-z^{4}a^{-6}+47z^{4}-a^{5}z^{3}+7a^{3}z^{3}+20az^{3}+22z^{3}a^{-1}+14z^{3}a^{-3}+4z^{3}a^{-5}+a^{4}z^{2}-5a^{2}z^{2}-12z^{2}a^{-2}-2z^{2}a^{-4}-16z^{2}-2a^{3}z-5az-5za^{-1}-za^{-3}+za^{-5}+a^{2}-a^{-2}-a^{-4}+2$
The A2 invariant	$-q^{14}+2q^{12}-3q^{10}+2q^{8}+q^{6}-4q^{4}+5q^{2}-4+4q^{-2}+q^{-4}+5q^{-8}-4q^{-10}+q^{-12}-q^{-14}-2q^{-16}+q^{-18}$
The G2 invariant	$q^{80}-3q^{78}+7q^{76}-13q^{74}+16q^{72}-16q^{70}+7q^{68}+16q^{66}-45q^{64}+83q^{62}-114q^{60}+115q^{58}-80q^{56}-9q^{54}+139q^{52}-278q^{50}+389q^{48}-410q^{46}+295q^{44}-41q^{42}-306q^{40}+643q^{38}-839q^{36}+789q^{34}-473q^{32}-53q^{30}+605q^{28}-976q^{26}+1019q^{24}-679q^{22}+96q^{20}+491q^{18}-837q^{16}+777q^{14}-340q^{12}-279q^{10}+797q^{8}-952q^{6}+644q^{4}+28q^{2}-796+1345q^{-2}-1425q^{-4}+976q^{-6}-156q^{-8}-748q^{-10}+1412q^{-12}-1590q^{-14}+1240q^{-16}-491q^{-18}-358q^{-20}+997q^{-22}-1192q^{-24}+906q^{-26}-281q^{-28}-387q^{-30}+813q^{-32}-819q^{-34}+412q^{-36}+228q^{-38}-798q^{-40}+1054q^{-42}-878q^{-44}+334q^{-46}+339q^{-48}-892q^{-50}+1112q^{-52}-949q^{-54}+500q^{-56}+44q^{-58}-492q^{-60}+703q^{-62}-662q^{-64}+443q^{-66}-155q^{-68}-92q^{-70}+226q^{-72}-253q^{-74}+196q^{-76}-106q^{-78}+32q^{-80}+21q^{-82}-39q^{-84}+35q^{-86}-24q^{-88}+11q^{-90}-4q^{-92}+q^{-94}$

Further Quantum Invariants

K11a72 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["K11a72"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$t^{4}-6t^{3}+18t^{2}-32t+39-32t^{-1}+18t^{-2}-6t^{-3}+t^{-4}$

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]=

{ 153, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$q^{6}-5q^{5}+10q^{4}-16q^{3}+22q^{2}-24q+25-21q^{-1}+15q^{-2}-9q^{-3}+4q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$z^{8}-a^{2}z^{6}-2z^{6}a^{-2}+5z^{6}-3a^{2}z^{4}-6z^{4}a^{-2}+z^{4}a^{-4}+10z^{4}-3a^{2}z^{2}-4z^{2}a^{-2}+z^{2}a^{-4}+8z^{2}-a^{2}+a^{-2}-a^{-4}+2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$2z^{10}a^{-2}+2z^{10}+7az^{9}+14z^{9}a^{-1}+7z^{9}a^{-3}+10a^{2}z^{8}+19z^{8}a^{-2}+9z^{8}a^{-4}+20z^{8}+8a^{3}z^{7}+az^{7}-16z^{7}a^{-1}-4z^{7}a^{-3}+5z^{7}a^{-5}+4a^{4}z^{6}-15a^{2}z^{6}-55z^{6}a^{-2}-20z^{6}a^{-4}+z^{6}a^{-6}-53z^{6}+a^{5}z^{5}-12a^{3}z^{5}-20az^{5}-15z^{5}a^{-1}-18z^{5}a^{-3}-10z^{5}a^{-5}-5a^{4}z^{4}+11a^{2}z^{4}+44z^{4}a^{-2}+12z^{4}a^{-4}-z^{4}a^{-6}+47z^{4}-a^{5}z^{3}+7a^{3}z^{3}+20az^{3}+22z^{3}a^{-1}+14z^{3}a^{-3}+4z^{3}a^{-5}+a^{4}z^{2}-5a^{2}z^{2}-12z^{2}a^{-2}-2z^{2}a^{-4}-16z^{2}-2a^{3}z-5az-5za^{-1}-za^{-3}+za^{-5}+a^{2}-a^{-2}-a^{-4}+2$

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 $8$ $8$ $32$ ${\frac {124}{3}}$ $-{\frac {4}{3}}$ $64$ ${\frac {368}{3}}$ ${\frac {128}{3}}$ $8$ ${\frac {256}{3}}$ $32$ ${\frac {992}{3}}$ $-{\frac {32}{3}}$ ${\frac {6751}{15}}$ $-{\frac {164}{15}}$ ${\frac {7684}{45}}$ $-{\frac {31}{9}}$ ${\frac {271}{15}}$

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=$0 is the signature of K11a72. 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 13                       1 1 11                     4   -4 9                   6 1   5 7                 10 4     -6 5               12 6       6 3             12 10         -2 1           13 12           1 -1         9 13             4 -3       6 12               -6 -5     3 9                 6 -7   1 6                   -5 -9   3                     3 -11 1                       -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=0$ ${\mathbb {Z} }^{13}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{13}$ $r=1$ ${\mathbb {Z} }^{12}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=2$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{12}$ ${\mathbb {Z} }^{12}$ $r=3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{10}$ ${\mathbb {Z} }^{10}$ $r=4$ ${\mathbb {Z} }^{4}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=5$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{4}$ ${\mathbb {Z} }^{4}$ $r=6$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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, 72]]
Out[2]=	11
In[3]:=	PD[Knot[11, Alternating, 72]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[12, 5, 13, 6], X[14, 8, 15, 7], X[2, 10, 3, 9], X[22, 11, 1, 12], X[18, 14, 19, 13], X[20, 15, 21, 16], X[8, 18, 9, 17], X[6, 19, 7, 20], X[16, 21, 17, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 72]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -10, 4, -9, 5, -2, 6, -3, 7, -4, 8, -11, 9, -7, 10, -8, 11, -6]
In[5]:=	BR[Knot[11, Alternating, 72]]
Out[5]=	BR[Knot[11, Alternating, 72]]
In[6]:=	alex = Alexander[Knot[11, Alternating, 72]][t]
Out[6]=	-4 6 18 32 2 3 4 39 + t - -- + -- - -- - 32 t + 18 t - 6 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 72]][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, 72]}
In[9]:=	{KnotDet[Knot[11, Alternating, 72]], KnotSignature[Knot[11, Alternating, 72]]}
Out[9]=	{153, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 72]][q]
Out[10]=	-5 4 9 15 21 2 3 4 5 6 25 - q + -- - -- + -- - -- - 24 q + 22 q - 16 q + 10 q - 5 q + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 72]}
In[12]:=	A2Invariant[Knot[11, Alternating, 72]][q]
Out[12]=	-14 2 3 2 -6 4 5 2 4 8 10 -4 - q + --- - --- + -- + q - -- + -- + 4 q + q + 5 q - 4 q + 12 10 8 4 2 q q q q q 12 14 16 18 q - q - 2 q + q
In[13]:=	Kauffman[Knot[11, Alternating, 72]][a, z]
Out[13]=	2 -4 -2 2 z z 5 z 3 2 2 z 2 - a - a + a + -- - -- - --- - 5 a z - 2 a z - 16 z - ---- - 5 3 a 4 a a a 2 3 3 3 12 z 2 2 4 2 4 z 14 z 22 z 3 3 3 ----- - 5 a z + a z + ---- + ----- + ----- + 20 a z + 7 a z - 2 5 3 a a a a 4 4 4 5 5 3 4 z 12 z 44 z 2 4 4 4 10 z a z + 47 z - -- + ----- + ----- + 11 a z - 5 a z - ----- - 6 4 2 5 a a a a 5 5 6 6 18 z 15 z 5 3 5 5 5 6 z 20 z ----- - ----- - 20 a z - 12 a z + a z - 53 z + -- - ----- - 3 a 6 4 a a a 6 7 7 7 55 z 2 6 4 6 5 z 4 z 16 z 7 3 7 ----- - 15 a z + 4 a z + ---- - ---- - ----- + a z + 8 a z + 2 5 3 a a a a 8 8 9 9 8 9 z 19 z 2 8 7 z 14 z 9 10 20 z + ---- + ----- + 10 a z + ---- + ----- + 7 a z + 2 z + 4 2 3 a a a a 10 2 z ----- 2 a
In[14]:=	{Vassiliev[2][Knot[11, Alternating, 72]], Vassiliev[3][Knot[11, Alternating, 72]]}
Out[14]=	{0, 1}
In[15]:=	Kh[Knot[11, Alternating, 72]][q, t]
Out[15]=	13 1 3 1 6 3 9 6 -- + 13 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t 12 9 3 3 2 5 2 5 3 ---- + --- + 12 q t + 12 q t + 10 q t + 12 q t + 6 q t + 3 q t q t 7 3 7 4 9 4 9 5 11 5 13 6 10 q t + 4 q t + 6 q t + q t + 4 q t + q t