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

K11n153 K11n155

Knotscape

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DrawMorseLink

PD Presentation:

X6271 X8394 X12,6,13,5 X20,8,21,7 X9,17,10,16 X11,19,12,18 X22,13,1,14 X4,16,5,15 X17,11,18,10 X2,19,3,20 X14,21,15,22

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-3 - 7t-2 + 19t-1 - 25 + 19t - 7t2 + t3

Conway Polynomial:

1 - z4 + z6

Other knots with the same Alexander/Conway Polynomial:

{1044, ...}

Determinant and Signature:

{79, 2}

Jones Polynomial:

2q-1 - 5 + 9q - 12q2 + 14q3 - 13q4 + 11q5 - 8q6 + 4q7 - q8

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

{...}

A2 (sl(3)) Invariant:

2q-4 - 1 + 3q2 - 3q4 + q6 - q10 + 3q12 - 2q14 + 3q16 - q18 - 2q20 + 2q22 - q24

HOMFLY-PT Polynomial:

- a-6 - a-6z2 - a-6z4 + 3a-4 + 5a-4z2 + 3a-4z4 + a-4z6 - 3a-2 - 6a-2z2 - 3a-2z4 + 2 + 2z2

Kauffman Polynomial:

- a-9z3 + a-9z5 + a-8z2 - 6a-8z4 + 4a-8z6 - a-7z + 5a-7z3 - 13a-7z5 + 7a-7z7 + a-6 - 2a-6z2 + 3a-6z4 - 9a-6z6 + 6a-6z8 - a-5z + 9a-5z3 - 16a-5z5 + 6a-5z7 + 2a-5z9 + 3a-4 - 11a-4z2 + 20a-4z4 - 19a-4z6 + 9a-4z8 + a-3z - 2a-3z3 + a-3z5 + 2a-3z9 + 3a-2 - 13a-2z2 + 14a-2z4 - 6a-2z6 + 3a-2z8 + a-1z - 5a-1z3 + 3a-1z5 + a-1z7 + 2 - 5z2 + 3z4

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

{0, 1}

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=2 is the signature of 11154. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 17                   1 j = 15                 3   j = 13               5 1   j = 11             6 3     j = 9           7 5       j = 7         7 6         j = 5       5 7           j = 3     4 7             j = 1   2 6               j = -1   3                 j = -3 2

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, 154]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 154]]
Out[3]=	PD[X[6, 2, 7, 1], X[8, 3, 9, 4], X[12, 6, 13, 5], X[20, 8, 21, 7], > X[9, 17, 10, 16], X[11, 19, 12, 18], X[22, 13, 1, 14], X[4, 16, 5, 15], > X[17, 11, 18, 10], X[2, 19, 3, 20], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 154]]
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, 154]]
Out[5]=	DTCode[6, 8, 12, 20, -16, -18, 22, 4, -10, 2, 14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 154]][t]
Out[6]=	-3 7 19 2 3 -25 + t - -- + -- + 19 t - 7 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 154]][z]
Out[7]=	4 6 1 - z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 44], Knot[11, NonAlternating, 154]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 154]], KnotSignature[Knot[11, NonAlternating, 154]]}
Out[9]=	{79, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 154]][q]
Out[10]=	2 2 3 4 5 6 7 8 -5 + - + 9 q - 12 q + 14 q - 13 q + 11 q - 8 q + 4 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 154]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 154]][q]
Out[12]=	2 2 4 6 10 12 14 16 18 20 -1 + -- + 3 q - 3 q + q - q + 3 q - 2 q + 3 q - q - 2 q + 4 q 22 24 > 2 q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 154]][a, z]
Out[13]=	2 2 2 4 4 4 6 -6 3 3 2 z 5 z 6 z z 3 z 3 z z 2 - a + -- - -- + 2 z - -- + ---- - ---- - -- + ---- - ---- + -- 4 2 6 4 2 6 4 2 4 a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 154]][a, z]
Out[14]=	2 2 2 2 3 -6 3 3 z z z z 2 z 2 z 11 z 13 z z 2 + a + -- + -- - -- - -- + -- + - - 5 z + -- - ---- - ----- - ----- - -- + 4 2 7 5 3 a 8 6 4 2 9 a a a a a a a a a a 3 3 3 3 4 4 4 4 5 5 z 9 z 2 z 5 z 4 6 z 3 z 20 z 14 z z > ---- + ---- - ---- - ---- + 3 z - ---- + ---- + ----- + ----- + -- - 7 5 3 a 8 6 4 2 9 a a a a a a a a 5 5 5 5 6 6 6 6 7 7 7 13 z 16 z z 3 z 4 z 9 z 19 z 6 z 7 z 6 z z > ----- - ----- + -- + ---- + ---- - ---- - ----- - ---- + ---- + ---- + -- + 7 5 3 a 8 6 4 2 7 5 a a a a a a a a a a 8 8 8 9 9 6 z 9 z 3 z 2 z 2 z > ---- + ---- + ---- + ---- + ---- 6 4 2 5 3 a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 154]], Vassiliev[3][Knot[11, NonAlternating, 154]]}
Out[15]=	{0, 1}
In[16]:=	Kh[Knot[11, NonAlternating, 154]][q, t]
Out[16]=	3 2 3 2 q 3 5 5 2 7 2 6 q + 4 q + ----- + --- + --- + 7 q t + 5 q t + 7 q t + 7 q t + 3 2 q t t q t 7 3 9 3 9 4 11 4 11 5 13 5 13 6 > 6 q t + 7 q t + 5 q t + 6 q t + 3 q t + 5 q t + q t + 15 6 17 7 > 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n154

K11n153 K11n155