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

K11n157 K11n159

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X10,3,11,4 X12,6,13,5 X18,8,19,7 X20,10,21,9 X16,11,17,12 X13,1,14,22 X4,16,5,15 X2,17,3,18 X8,20,9,19 X21,15,22,14

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

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

Conway Polynomial:

1 + z2 + 3z4 + 4z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{45, 4}

Jones Polynomial:

- q-1 + 3 - 4q + 7q2 - 7q3 + 7q4 - 7q5 + 5q6 - 3q7 + q8

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

{...}

A2 (sl(3)) Invariant:

- q-2 + 1 + 2q4 + 2q6 + q8 + 2q10 - 3q12 + q14 - 2q16 - q22 + q24 - q26 + q28

HOMFLY-PT Polynomial:

a-8 + a-8z2 - 3a-6 - 7a-6z2 - 5a-6z4 - a-6z6 + 2a-4 + 10a-4z2 + 12a-4z4 + 6a-4z6 + a-4z8 + a-2 - 3a-2z2 - 4a-2z4 - a-2z6

Kauffman Polynomial:

a-10z2 - a-9z + 3a-9z3 + a-8 - a-8z2 + a-8z4 + a-8z6 - a-7z + 5a-7z3 - 7a-7z5 + 3a-7z7 + 3a-6 - 12a-6z2 + 17a-6z4 - 14a-6z6 + 4a-6z8 + 2a-5z - 2a-5z5 - 4a-5z7 + 2a-5z9 + 2a-4 - 17a-4z2 + 35a-4z4 - 29a-4z6 + 7a-4z8 + 2a-3z + 2a-3z3 + a-3z5 - 6a-3z7 + 2a-3z9 - a-2 - 7a-2z2 + 19a-2z4 - 14a-2z6 + 3a-2z8 + 4a-1z3 - 4a-1z5 + a-1z7

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

{1, 0}

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=4 is the signature of 11158. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 17                   1 j = 15                 2   j = 13               3 1   j = 11             4 2     j = 9           3 3       j = 7         4 4         j = 5       3 3           j = 3     2 5             j = 1   1 2               j = -1   2                 j = -3 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, 158]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 158]]
Out[3]=	PD[X[6, 2, 7, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[18, 8, 19, 7], > X[20, 10, 21, 9], X[16, 11, 17, 12], X[13, 1, 14, 22], X[4, 16, 5, 15], > X[2, 17, 3, 18], X[8, 20, 9, 19], X[21, 15, 22, 14]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 158]]
Out[4]=	GaussCode[1, -9, 2, -8, 3, -1, 4, -10, 5, -2, 6, -3, -7, 11, 8, -6, 9, -4, 10, > -5, -11, 7]
In[5]:=	DTCode[Knot[11, NonAlternating, 158]]
Out[5]=	DTCode[6, 10, 12, 18, 20, 16, -22, 4, 2, 8, -14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 158]][t]
Out[6]=	-4 4 7 7 2 3 4 7 + t - -- + -- - - - 7 t + 7 t - 4 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 158]][z]
Out[7]=	2 4 6 8 1 + z + 3 z + 4 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 158]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 158]], KnotSignature[Knot[11, NonAlternating, 158]]}
Out[9]=	{45, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 158]][q]
Out[10]=	1 2 3 4 5 6 7 8 3 - - - 4 q + 7 q - 7 q + 7 q - 7 q + 5 q - 3 q + q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 158]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 158]][q]
Out[12]=	-2 4 6 8 10 12 14 16 22 24 26 28 1 - q + 2 q + 2 q + q + 2 q - 3 q + q - 2 q - q + q - q + q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 158]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 -8 3 2 -2 z 7 z 10 z 3 z 5 z 12 z 4 z z a - -- + -- + a + -- - ---- + ----- - ---- - ---- + ----- - ---- - -- + 6 4 8 6 4 2 6 4 2 6 a a a a a a a a a a 6 6 8 6 z z z > ---- - -- + -- 4 2 4 a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 158]][a, z]
Out[14]=	2 2 2 2 2 -8 3 2 -2 z z 2 z 2 z z z 12 z 17 z 7 z a + -- + -- - a - -- - -- + --- + --- + --- - -- - ----- - ----- - ---- + 6 4 9 7 5 3 10 8 6 4 2 a a a a a a a a a a a 3 3 3 3 4 4 4 4 5 5 5 3 z 5 z 2 z 4 z z 17 z 35 z 19 z 7 z 2 z z > ---- + ---- + ---- + ---- + -- + ----- + ----- + ----- - ---- - ---- + -- - 9 7 3 a 8 6 4 2 7 5 3 a a a a a a a a a a 5 6 6 6 6 7 7 7 7 8 8 4 z z 14 z 29 z 14 z 3 z 4 z 6 z z 4 z 7 z > ---- + -- - ----- - ----- - ----- + ---- - ---- - ---- + -- + ---- + ---- + a 8 6 4 2 7 5 3 a 6 4 a a a a a a a a a 8 9 9 3 z 2 z 2 z > ---- + ---- + ---- 2 5 3 a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 158]], Vassiliev[3][Knot[11, NonAlternating, 158]]}
Out[15]=	{1, 0}
In[16]:=	Kh[Knot[11, NonAlternating, 158]][q, t]
Out[16]=	3 3 5 1 2 q 2 q 2 q 5 7 7 2 5 q + 3 q + ----- + ---- + -- + --- + ---- + 3 q t + 4 q t + 4 q t + 3 3 2 2 t t q t q t t 9 2 9 3 11 3 11 4 13 4 13 5 15 5 > 3 q t + 3 q t + 4 q t + 2 q t + 3 q t + q t + 2 q t + 17 6 > q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n158

K11n157 K11n159