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

K11n51 K11n53

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X5,14,6,15 X2837 X16,10,17,9 X18,11,19,12 X13,6,14,7 X22,16,1,15 X20,18,21,17 X10,19,11,20 X12,22,13,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

t-3 - 6t-2 + 14t-1 - 17 + 14t - 6t2 + t3

Conway Polynomial:

1 - z2 + z6

Other knots with the same Alexander/Conway Polynomial:

{932, K11n124, ...}

Determinant and Signature:

{59, 2}

Jones Polynomial:

2q-1 - 4 + 7q - 9q2 + 10q3 - 10q4 + 8q5 - 5q6 + 3q7 - q8

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

{...}

A2 (sl(3)) Invariant:

2q-4 + q-2 + 2q2 - 3q4 - q8 - q10 + 2q12 - q14 + 3q16 - q20 + q22 - q24

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- 2a-9z3 + a-9z5 + 3a-8z2 - 7a-8z4 + 3a-8z6 - 2a-7z + 6a-7z3 - 9a-7z5 + 4a-7z7 + a-6 + a-6z2 - 4a-6z6 + 3a-6z8 - 5a-5z + 15a-5z3 - 13a-5z5 + 4a-5z7 + a-5z9 + 4a-4 - 13a-4z2 + 18a-4z4 - 12a-4z6 + 5a-4z8 - 3a-3z + 4a-3z3 - 2a-3z5 + a-3z7 + a-3z9 + 5a-2 - 17a-2z2 + 14a-2z4 - 5a-2z6 + 2a-2z8 - 3a-1z3 + a-1z5 + a-1z7 + 3 - 6z2 + 3z4

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=2 is the signature of 1152. 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                 2   j = 13               3 1   j = 11             5 2     j = 9           5 3       j = 7         5 5         j = 5       4 5           j = 3     3 5             j = 1   2 5               j = -1   2                 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, 52]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 52]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 14, 6, 15], X[2, 8, 3, 7], > X[16, 10, 17, 9], X[18, 11, 19, 12], X[13, 6, 14, 7], X[22, 16, 1, 15], > X[20, 18, 21, 17], X[10, 19, 11, 20], X[12, 22, 13, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 52]]
Out[4]=	GaussCode[1, -4, 2, -1, -3, 7, 4, -2, 5, -10, 6, -11, -7, 3, 8, -5, 9, -6, 10, > -9, 11, -8]
In[5]:=	DTCode[Knot[11, NonAlternating, 52]]
Out[5]=	DTCode[4, 8, -14, 2, 16, 18, -6, 22, 20, 10, 12]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 52]][t]
Out[6]=	-3 6 14 2 3 -17 + t - -- + -- + 14 t - 6 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 52]][z]
Out[7]=	2 6 1 - z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 32], Knot[11, NonAlternating, 52], Knot[11, NonAlternating, 124]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 52]], KnotSignature[Knot[11, NonAlternating, 52]]}
Out[9]=	{59, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 52]][q]
Out[10]=	2 2 3 4 5 6 7 8 -4 + - + 7 q - 9 q + 10 q - 10 q + 8 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, 52]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 52]][q]
Out[12]=	2 -2 2 4 8 10 12 14 16 20 22 24 -- + q + 2 q - 3 q - q - q + 2 q - q + 3 q - q + q - q 4 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 52]][a, z]
Out[13]=	2 2 2 4 4 4 6 -6 4 5 2 2 z 7 z 8 z z 4 z 3 z z 3 - 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, 52]][a, z]
Out[14]=	2 2 2 2 3 -6 4 5 2 z 5 z 3 z 2 3 z z 13 z 17 z 2 z 3 + a + -- + -- - --- - --- - --- - 6 z + ---- + -- - ----- - ----- - ---- + 4 2 7 5 3 8 6 4 2 9 a a a a a a a a a a 3 3 3 3 4 4 4 5 5 6 z 15 z 4 z 3 z 4 7 z 18 z 14 z z 9 z > ---- + ----- + ---- - ---- + 3 z - ---- + ----- + ----- + -- - ---- - 7 5 3 a 8 4 2 9 7 a a a a a a a a 5 5 5 6 6 6 6 7 7 7 7 13 z 2 z z 3 z 4 z 12 z 5 z 4 z 4 z z z > ----- - ---- + -- + ---- - ---- - ----- - ---- + ---- + ---- + -- + -- + 5 3 a 8 6 4 2 7 5 3 a a a a a a a a a a 8 8 8 9 9 3 z 5 z 2 z z z > ---- + ---- + ---- + -- + -- 6 4 2 5 3 a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 52]], Vassiliev[3][Knot[11, NonAlternating, 52]]}
Out[15]=	{-1, 0}
In[16]:=	Kh[Knot[11, NonAlternating, 52]][q, t]
Out[16]=	3 2 2 2 q 3 5 5 2 7 2 5 q + 3 q + ----- + --- + --- + 5 q t + 4 q t + 5 q t + 5 q t + 3 2 q t t q t 7 3 9 3 9 4 11 4 11 5 13 5 13 6 > 5 q t + 5 q t + 3 q t + 5 q t + 2 q t + 3 q t + q t + 15 6 17 7 > 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n52

K11n51 K11n53