© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11n152 K11n154

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X8394 X12,6,13,5 X20,8,21,7 X16,9,17,10 X18,11,19,12 X13,1,14,22 X4,16,5,15 X10,17,11,18 X2,19,3,20 X21,15,22,14

Gauss Code:

{1, -10, 2, -8, 3, -1, 4, -2, 5, -9, 6, -3, -7, 11, 8, -5, 9, -6, 10, -4, -11, 7}

DT (Dowker-Thistlethwaite) Code:

6 8 12 20 16 18 -22 4 10 2 -14

Alexander Polynomial:

t-4 - 4t-3 + 7t-2 - 10t-1 + 13 - 10t + 7t2 - 4t3 + t4

Conway Polynomial:

1 - 2z2 + 3z4 + 4z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{57, 0}

Jones Polynomial:

- q-5 + 3q-4 - 5q-3 + 8q-2 - 9q-1 + 10 - 9q + 6q2 - 4q3 + 2q4

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

- q-14 + q-12 - q-10 + 2q-8 + q-6 + 3q-2 - 2 + 2q2 - 3q4 - q6 - q10 + 2q12 + q16

HOMFLY-PT Polynomial:

2a-4 + a-4z2 - 4a-2 - 8a-2z2 - 5a-2z4 - a-2z6 + 3 + 9z2 + 12z4 + 6z6 + z8 - 4a2z2 - 4a2z4 - a2z6

Kauffman Polynomial:

2a-4 - 7a-4z2 + 3a-4z4 + 3a-3z - 6a-3z3 + a-3z5 + a-3z7 + 4a-2 - 21a-2z2 + 25a-2z4 - 12a-2z6 + 3a-2z8 + 3a-1z - 7a-1z3 + 11a-1z5 - 6a-1z7 + 2a-1z9 + 3 - 21z2 + 41z4 - 26z6 + 7z8 - az + 5az3 - 3az7 + 2az9 - 5a2z2 + 12a2z4 - 11a2z6 + 4a2z8 - a3z + 4a3z3 - 9a3z5 + 4a3z7 + 2a4z2 - 7a4z4 + 3a4z6 - 2a5z3 + a5z5

V2 and V3, the type 2 and 3 Vassiliev invariants:

{-2, -2}

Khovanov Homology: (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 11153. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 9                   2 j = 7                 2   j = 5               4 2   j = 3             5 2     j = 1           5 4       j = -1         5 6         j = -3       3 4           j = -5     2 5             j = -7   1 3               j = -9   2                 j = -11 1

Computer Talk. The data above can be recomputed by Mathematica using the package KnotTheory`. Following setup, the sample Mathematica session below reproduces most of the above data (Mathematica system prompts in blue, human input in red, Mathematica output in black):

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 30, 2005, 10:15:35)...
In[2]:=	Show[DrawMorseLink[Knot[11, NonAlternating, 153]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 153]]
Out[3]=	PD[X[6, 2, 7, 1], X[8, 3, 9, 4], X[12, 6, 13, 5], X[20, 8, 21, 7], > X[16, 9, 17, 10], X[18, 11, 19, 12], X[13, 1, 14, 22], X[4, 16, 5, 15], > X[10, 17, 11, 18], X[2, 19, 3, 20], X[21, 15, 22, 14]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 153]]
Out[4]=	GaussCode[1, -10, 2, -8, 3, -1, 4, -2, 5, -9, 6, -3, -7, 11, 8, -5, 9, -6, 10, > -4, -11, 7]
In[5]:=	DTCode[Knot[11, NonAlternating, 153]]
Out[5]=	DTCode[6, 8, 12, 20, 16, 18, -22, 4, 10, 2, -14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 153]][t]
Out[6]=	-4 4 7 10 2 3 4 13 + t - -- + -- - -- - 10 t + 7 t - 4 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 153]][z]
Out[7]=	2 4 6 8 1 - 2 z + 3 z + 4 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 153]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 153]], KnotSignature[Knot[11, NonAlternating, 153]]}
Out[9]=	{57, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 153]][q]
Out[10]=	-5 3 5 8 9 2 3 4 10 - q + -- - -- + -- - - - 9 q + 6 q - 4 q + 2 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, NonAlternating, 153]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 153]][q]
Out[12]=	-14 -12 -10 2 -6 3 2 4 6 10 12 16 -2 - q + q - q + -- + q + -- + 2 q - 3 q - q - q + 2 q + q 8 2 q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 153]][a, z]
Out[13]=	2 2 4 6 2 4 2 z 8 z 2 2 4 5 z 2 4 6 z 3 + -- - -- + 9 z + -- - ---- - 4 a z + 12 z - ---- - 4 a z + 6 z - -- - 4 2 4 2 2 2 a a a a a a 2 6 8 > a z + z
In[14]:=	Kauffman[Knot[11, NonAlternating, 153]][a, z]
Out[14]=	2 2 2 4 3 z 3 z 3 2 7 z 21 z 2 2 3 + -- + -- + --- + --- - a z - a z - 21 z - ---- - ----- - 5 a z + 4 2 3 a 4 2 a a a a a 3 3 4 4 4 2 6 z 7 z 3 3 3 5 3 4 3 z 25 z > 2 a z - ---- - ---- + 5 a z + 4 a z - 2 a z + 41 z + ---- + ----- + 3 a 4 2 a a a 5 5 6 2 4 4 4 z 11 z 3 5 5 5 6 12 z > 12 a z - 7 a z + -- + ----- - 9 a z + a z - 26 z - ----- - 3 a 2 a a 7 7 8 2 6 4 6 z 6 z 7 3 7 8 3 z 2 8 > 11 a z + 3 a z + -- - ---- - 3 a z + 4 a z + 7 z + ---- + 4 a z + 3 a 2 a a 9 2 z 9 > ---- + 2 a z a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 153]], Vassiliev[3][Knot[11, NonAlternating, 153]]}
Out[15]=	{-2, -2}
In[16]:=	Kh[Knot[11, NonAlternating, 153]][q, t]
Out[16]=	6 1 2 1 3 2 5 3 4 5 - + 5 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + --- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t q t 3 3 2 5 2 5 3 7 3 9 4 > 4 q t + 5 q t + 2 q t + 4 q t + 2 q t + 2 q t + 2 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n153

K11n152 K11n154