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

K11a153 K11a155

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X18,6,19,5 X12,8,13,7 X2,10,3,9 X8,12,9,11 X22,13,1,14 X20,15,21,16 X6,18,7,17 X16,19,17,20 X14,21,15,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 18 12 2 8 22 20 6 16 14

Alexander Polynomial:

- 4t-2 + 17t-1 - 25 + 17t - 4t2

Conway Polynomial:

1 + z2 - 4z4

Other knots with the same Alexander/Conway Polynomial:

{1030, ...}

Determinant and Signature:

{67, 2}

Jones Polynomial:

q-3 - 2q-2 + 4q-1 - 7 + 9q - 10q2 + 11q3 - 9q4 + 7q5 - 4q6 + 2q7 - q8

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

{...}

A2 (sl(3)) Invariant:

q-10 + 2q-4 - 2q-2 - 1 + q2 - 2q4 + 2q6 + q10 + q12 - q14 + 3q16 - q20 + q22 - q24 - q26

HOMFLY-PT Polynomial:

- a-8 + a-6 + 2a-6z2 + a-4 + a-4z2 - a-4z4 - 2a-2z2 - 2a-2z4 - 1 - z2 - z4 + a2 + a2z2

Kauffman Polynomial:

2a-9z - 3a-9z3 + a-9z5 - a-8 + 3a-8z2 - 5a-8z4 + 2a-8z6 + 2a-7z - 3a-7z3 - 2a-7z5 + 2a-7z7 - a-6 + 2a-6z2 - a-6z4 - 2a-6z6 + 2a-6z8 + a-5z - 5a-5z3 + 9a-5z5 - 5a-5z7 + 2a-5z9 + a-4 - 3a-4z2 + 5a-4z4 - a-4z8 + a-4z10 + a-3z - 11a-3z3 + 22a-3z5 - 14a-3z7 + 4a-3z9 - a-2z2 + 2a-2z4 - a-2z6 - a-2z8 + a-2z10 - 2a-1z + 3a-1z5 - 5a-1z7 + 2a-1z9 - 1 + 5z2 - 3z4 - 4z6 + 2z8 - 2az + 6az3 - 7az5 + 2az7 - a2 + 4a2z2 - 4a2z4 + a2z6

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

{1, 5}

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 = -4 r = -3 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                     1   j = 13                   3 1   j = 11                 4 1     j = 9               5 3       j = 7             6 4         j = 5           4 5           j = 3         5 6             j = 1       3 5               j = -1     1 4                 j = -3   1 3                   j = -5   1                     j = -7 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, Alternating, 154]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 154]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[18, 6, 19, 5], X[12, 8, 13, 7], > X[2, 10, 3, 9], X[8, 12, 9, 11], X[22, 13, 1, 14], X[20, 15, 21, 16], > X[6, 18, 7, 17], X[16, 19, 17, 20], X[14, 21, 15, 22]]
In[4]:=	GaussCode[Knot[11, Alternating, 154]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, 4, -6, 5, -2, 6, -4, 7, -11, 8, -10, 9, -3, 10, > -8, 11, -7]
In[5]:=	DTCode[Knot[11, Alternating, 154]]
Out[5]=	DTCode[4, 10, 18, 12, 2, 8, 22, 20, 6, 16, 14]
In[6]:=	alex = Alexander[Knot[11, Alternating, 154]][t]
Out[6]=	4 17 2 -25 - -- + -- + 17 t - 4 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 154]][z]
Out[7]=	2 4 1 + z - 4 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 30], Knot[11, Alternating, 154]}
In[9]:=	{KnotDet[Knot[11, Alternating, 154]], KnotSignature[Knot[11, Alternating, 154]]}
Out[9]=	{67, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 154]][q]
Out[10]=	-3 2 4 2 3 4 5 6 7 8 -7 + q - -- + - + 9 q - 10 q + 11 q - 9 q + 7 q - 4 q + 2 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 154]}
In[12]:=	A2Invariant[Knot[11, Alternating, 154]][q]
Out[12]=	-10 2 2 2 4 6 10 12 14 16 20 22 -1 + q + -- - -- + q - 2 q + 2 q + q + q - q + 3 q - q + q - 4 2 q q 24 26 > q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 154]][a, z]
Out[13]=	2 2 2 4 4 -8 -6 -4 2 2 2 z z 2 z 2 2 4 z 2 z -1 - a + a + a + a - z + ---- + -- - ---- + a z - z - -- - ---- 6 4 2 4 2 a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 154]][a, z]
Out[14]=	2 -8 -6 -4 2 2 z 2 z z z 2 z 2 3 z -1 - a - a + a - a + --- + --- + -- + -- - --- - 2 a z + 5 z + ---- + 9 7 5 3 a 8 a a a a a 2 2 2 3 3 3 3 2 z 3 z z 2 2 3 z 3 z 5 z 11 z 3 4 > ---- - ---- - -- + 4 a z - ---- - ---- - ---- - ----- + 6 a z - 3 z - 6 4 2 9 7 5 3 a a a a a a a 4 4 4 4 5 5 5 5 5 5 z z 5 z 2 z 2 4 z 2 z 9 z 22 z 3 z > ---- - -- + ---- + ---- - 4 a z + -- - ---- + ---- + ----- + ---- - 8 6 4 2 9 7 5 3 a a a a a a a a a 6 6 6 7 7 7 7 5 6 2 z 2 z z 2 6 2 z 5 z 14 z 5 z > 7 a z - 4 z + ---- - ---- - -- + a z + ---- - ---- - ----- - ---- + 8 6 2 7 5 3 a a a a a a a 8 8 8 9 9 9 10 10 7 8 2 z z z 2 z 4 z 2 z z z > 2 a z + 2 z + ---- - -- - -- + ---- + ---- + ---- + --- + --- 6 4 2 5 3 a 4 2 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 154]], Vassiliev[3][Knot[11, Alternating, 154]]}
Out[15]=	{1, 5}
In[16]:=	Kh[Knot[11, Alternating, 154]][q, t]
Out[16]=	3 1 1 1 3 1 4 3 q 3 5 q + 5 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 6 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 > 4 q t + 5 q t + 6 q t + 4 q t + 5 q t + 3 q t + 4 q t + 11 5 13 5 13 6 15 6 17 7 > q t + 3 q t + q t + q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a154

K11a153 K11a155