K11a92

K11a91

K11a93

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	K11a92 Quick Notes

K11a92 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X10,3,11,4 X12,6,13,5 X16,7,17,8 X22,10,1,9 X2,11,3,12 X18,13,19,14 X20,15,21,16 X8,17,9,18 X14,19,15,20 X6,22,7,21
Gauss code	1, -6, 2, -1, 3, -11, 4, -9, 5, -2, 6, -3, 7, -10, 8, -4, 9, -7, 10, -8, 11, -5
Dowker-Thistlethwaite code	4 10 12 16 22 2 18 20 8 14 6
Conway Notation	[2121,3,2]

Three dimensional invariants

Symmetry type Reversible Unknotting number $\{1,2\}$ 3-genus 4 Bridge index 3 Super bridge index Missing Nakanishi index Missing Maximal Thurston-Bennequin number Data:K11a92/ThurstonBennequinNumber Hyperbolic Volume 14.2422 A-Polynomial See Data:K11a92/A-polynomial

[edit Notes for K11a92'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 K11a92's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{4}+5t^{3}-12t^{2}+21t-25+21t^{-1}-12t^{-2}+5t^{-3}-t^{-4}$
Conway polynomial	$-z^{8}-3z^{6}-2z^{4}+2z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 103, -2 }
Jones polynomial	$q^{3}-3q^{2}+6q-10+14q^{-1}-16q^{-2}+17q^{-3}-14q^{-4}+11q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$
HOMFLY-PT polynomial (db, data sources)	$-a^{2}z^{8}+2a^{4}z^{6}-6a^{2}z^{6}+z^{6}-a^{6}z^{4}+9a^{4}z^{4}-14a^{2}z^{4}+4z^{4}-3a^{6}z^{2}+14a^{4}z^{2}-14a^{2}z^{2}+5z^{2}-3a^{6}+7a^{4}-5a^{2}+2$
Kauffman polynomial (db, data sources)	$a^{4}z^{10}+a^{2}z^{10}+4a^{5}z^{9}+7a^{3}z^{9}+3az^{9}+6a^{6}z^{8}+10a^{4}z^{8}+8a^{2}z^{8}+4z^{8}+5a^{7}z^{7}-3a^{5}z^{7}-13a^{3}z^{7}-2az^{7}+3z^{7}a^{-1}+3a^{8}z^{6}-12a^{6}z^{6}-34a^{4}z^{6}-30a^{2}z^{6}+z^{6}a^{-2}-10z^{6}+a^{9}z^{5}-8a^{7}z^{5}-5a^{5}z^{5}+3a^{3}z^{5}-10az^{5}-9z^{5}a^{-1}-5a^{8}z^{4}+13a^{6}z^{4}+47a^{4}z^{4}+39a^{2}z^{4}-3z^{4}a^{-2}+7z^{4}-2a^{9}z^{3}+3a^{7}z^{3}+12a^{5}z^{3}+12a^{3}z^{3}+12az^{3}+7z^{3}a^{-1}+a^{8}z^{2}-10a^{6}z^{2}-27a^{4}z^{2}-23a^{2}z^{2}+2z^{2}a^{-2}-5z^{2}+a^{9}z-2a^{7}z-6a^{5}z-6a^{3}z-5az-2za^{-1}+3a^{6}+7a^{4}+5a^{2}+2$
The A2 invariant	$-q^{24}-q^{20}-2q^{18}+3q^{16}-q^{14}+3q^{12}+2q^{10}-q^{8}+3q^{6}-4q^{4}+2q^{2}-1-q^{-2}+2q^{-4}-q^{-6}+q^{-8}$
The G2 invariant	$q^{128}-2q^{126}+5q^{124}-8q^{122}+9q^{120}-7q^{118}+13q^{114}-28q^{112}+43q^{110}-53q^{108}+43q^{106}-19q^{104}-24q^{102}+81q^{100}-127q^{98}+153q^{96}-137q^{94}+64q^{92}+41q^{90}-162q^{88}+249q^{86}-269q^{84}+202q^{82}-65q^{80}-105q^{78}+239q^{76}-286q^{74}+225q^{72}-84q^{70}-83q^{68}+194q^{66}-206q^{64}+112q^{62}+54q^{60}-202q^{58}+279q^{56}-224q^{54}+60q^{52}+159q^{50}-339q^{48}+423q^{46}-355q^{44}+170q^{42}+81q^{40}-299q^{38}+419q^{36}-394q^{34}+241q^{32}-23q^{30}-180q^{28}+282q^{26}-261q^{24}+132q^{22}+42q^{20}-183q^{18}+218q^{16}-148q^{14}-7q^{12}+172q^{10}-275q^{8}+271q^{6}-166q^{4}-2q^{2}+163-265q^{-2}+280q^{-4}-203q^{-6}+83q^{-8}+43q^{-10}-134q^{-12}+170q^{-14}-152q^{-16}+102q^{-18}-37q^{-20}-18q^{-22}+51q^{-24}-61q^{-26}+52q^{-28}-32q^{-30}+15q^{-32}+q^{-34}-10q^{-36}+10q^{-38}-9q^{-40}+5q^{-42}-2q^{-44}+q^{-46}$

Further Quantum Invariants

K11a92 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["K11a92"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{4}+5t^{3}-12t^{2}+21t-25+21t^{-1}-12t^{-2}+5t^{-3}-t^{-4}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{8}-3z^{6}-2z^{4}+2z^{2}+1$

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]=

$q^{3}-3q^{2}+6q-10+14q^{-1}-16q^{-2}+17q^{-3}-14q^{-4}+11q^{-5}-7q^{-6}+3q^{-7}-q^{-8}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

$-a^{2}z^{8}+2a^{4}z^{6}-6a^{2}z^{6}+z^{6}-a^{6}z^{4}+9a^{4}z^{4}-14a^{2}z^{4}+4z^{4}-3a^{6}z^{2}+14a^{4}z^{2}-14a^{2}z^{2}+5z^{2}-3a^{6}+7a^{4}-5a^{2}+2$

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$a^{4}z^{10}+a^{2}z^{10}+4a^{5}z^{9}+7a^{3}z^{9}+3az^{9}+6a^{6}z^{8}+10a^{4}z^{8}+8a^{2}z^{8}+4z^{8}+5a^{7}z^{7}-3a^{5}z^{7}-13a^{3}z^{7}-2az^{7}+3z^{7}a^{-1}+3a^{8}z^{6}-12a^{6}z^{6}-34a^{4}z^{6}-30a^{2}z^{6}+z^{6}a^{-2}-10z^{6}+a^{9}z^{5}-8a^{7}z^{5}-5a^{5}z^{5}+3a^{3}z^{5}-10az^{5}-9z^{5}a^{-1}-5a^{8}z^{4}+13a^{6}z^{4}+47a^{4}z^{4}+39a^{2}z^{4}-3z^{4}a^{-2}+7z^{4}-2a^{9}z^{3}+3a^{7}z^{3}+12a^{5}z^{3}+12a^{3}z^{3}+12az^{3}+7z^{3}a^{-1}+a^{8}z^{2}-10a^{6}z^{2}-27a^{4}z^{2}-23a^{2}z^{2}+2z^{2}a^{-2}-5z^{2}+a^{9}z-2a^{7}z-6a^{5}z-6a^{3}z-5az-2za^{-1}+3a^{6}+7a^{4}+5a^{2}+2$

Vassiliev invariants

V2 and V3:

(2, -5)

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 $8$ $-40$ $32$ ${\frac {460}{3}}$ ${\frac {92}{3}}$ $-320$ $-{\frac {1936}{3}}$ $-{\frac {256}{3}}$ $-104$ ${\frac {256}{3}}$ $800$ ${\frac {3680}{3}}$ ${\frac {736}{3}}$ ${\frac {48391}{15}}$ ${\frac {1036}{15}}$ ${\frac {56884}{45}}$ ${\frac {377}{9}}$ ${\frac {2071}{15}}$

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 K11a92. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 χ 7                       1 1 5                     2   -2 3                   4 1   3 1                 6 2     -4 -1               8 4       4 -3             9 7         -2 -5           8 7           1 -7         6 9             3 -9       5 8               -3 -11     2 6                 4 -13   1 5                   -4 -15   2                     2 -17 1                       -1

Integral Khovanov Homology (db, data source)

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

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, 92]]
Out[2]=	11
In[3]:=	PD[Knot[11, Alternating, 92]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[16, 7, 17, 8], X[22, 10, 1, 9], X[2, 11, 3, 12], X[18, 13, 19, 14], X[20, 15, 21, 16], X[8, 17, 9, 18], X[14, 19, 15, 20], X[6, 22, 7, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 92]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -11, 4, -9, 5, -2, 6, -3, 7, -10, 8, -4, 9, -7, 10, -8, 11, -5]
In[5]:=	BR[Knot[11, Alternating, 92]]
Out[5]=	BR[Knot[11, Alternating, 92]]
In[6]:=	alex = Alexander[Knot[11, Alternating, 92]][t]
Out[6]=	-4 5 12 21 2 3 4 -25 - t + -- - -- + -- + 21 t - 12 t + 5 t - t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 92]][z]
Out[7]=	2 4 6 8 1 + 2 z - 2 z - 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 92]}
In[9]:=	{KnotDet[Knot[11, Alternating, 92]], KnotSignature[Knot[11, Alternating, 92]]}
Out[9]=	{103, -2}
In[10]:=	J=Jones[Knot[11, Alternating, 92]][q]
Out[10]=	-8 3 7 11 14 17 16 14 2 3 -10 - q + -- - -- + -- - -- + -- - -- + -- + 6 q - 3 q + q 7 6 5 4 3 2 q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 92]}
In[12]:=	A2Invariant[Knot[11, Alternating, 92]][q]
Out[12]=	-24 -20 2 3 -14 3 2 -8 3 4 2 -1 - q - q - --- + --- - q + --- + --- - q + -- - -- + -- - 18 16 12 10 6 4 2 q q q q q q q 2 4 6 8 q + 2 q - q + q
In[13]:=	Kauffman[Knot[11, Alternating, 92]][a, z]
Out[13]=	2 4 6 2 z 3 5 7 2 + 5 a + 7 a + 3 a - --- - 5 a z - 6 a z - 6 a z - 2 a z + a 2 3 9 2 2 z 2 2 4 2 6 2 8 2 7 z a z - 5 z + ---- - 23 a z - 27 a z - 10 a z + a z + ---- + 2 a a 4 3 3 3 5 3 7 3 9 3 4 3 z 12 a z + 12 a z + 12 a z + 3 a z - 2 a z + 7 z - ---- + 2 a 5 2 4 4 4 6 4 8 4 9 z 5 3 5 39 a z + 47 a z + 13 a z - 5 a z - ---- - 10 a z + 3 a z - a 6 5 5 7 5 9 5 6 z 2 6 4 6 5 a z - 8 a z + a z - 10 z + -- - 30 a z - 34 a z - 2 a 7 6 6 8 6 3 z 7 3 7 5 7 7 7 12 a z + 3 a z + ---- - 2 a z - 13 a z - 3 a z + 5 a z + a 8 2 8 4 8 6 8 9 3 9 5 9 4 z + 8 a z + 10 a z + 6 a z + 3 a z + 7 a z + 4 a z + 2 10 4 10 a z + a z
In[14]:=	{Vassiliev[2][Knot[11, Alternating, 92]], Vassiliev[3][Knot[11, Alternating, 92]]}
Out[14]=	{0, -5}
In[15]:=	Kh[Knot[11, Alternating, 92]][q, t]
Out[15]=	7 8 1 2 1 5 2 6 5 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 q q t q t q t q t q t q t q t 8 6 9 8 7 9 4 t 2 ----- + ----- + ----- + ----- + ---- + ---- + --- + 6 q t + 2 q t + 9 3 7 3 7 2 5 2 5 3 q q t q t q t q t q t q t 3 2 3 3 5 3 7 4 4 q t + q t + 2 q t + q t