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

K11a74 K11a76

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 12 14 18 2 6 22 20 8 16

Alexander Polynomial:

2t-3 - 10t-2 + 19t-1 - 21 + 19t - 10t2 + 2t3

Conway Polynomial:

1 - 3z2 + 2z4 + 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{83, 2}

Jones Polynomial:

q-3 - 2q-2 + 5q-1 - 8 + 11q - 13q2 + 13q3 - 12q4 + 9q5 - 5q6 + 3q7 - q8

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

{...}

A2 (sl(3)) Invariant:

q-10 + q-8 + 2q-4 - q-2 - 1 + q2 - 3q4 + 2q6 - q8 + q12 - 2q14 + 3q16 + q22 - q24

HOMFLY-PT Polynomial:

- 2a-6z2 - a-6z4 + a-4 + 3a-4z2 + 3a-4z4 + a-4z6 + 2a-2z4 + a-2z6 - 2 - 5z2 - 2z4 + 2a2 + a2z2

Kauffman Polynomial:

- 2a-9z3 + a-9z5 + 3a-8z2 - 7a-8z4 + 3a-8z6 - a-7z + 4a-7z3 - 8a-7z5 + 4a-7z7 + 2a-6z2 + 3a-6z4 - 7a-6z6 + 4a-6z8 - 2a-5z + 3a-5z3 + 4a-5z5 - 5a-5z7 + 3a-5z9 + a-4 - 8a-4z2 + 17a-4z4 - 11a-4z6 + 3a-4z8 + a-4z10 + 2a-3z - 13a-3z3 + 21a-3z5 - 14a-3z7 + 5a-3z9 - 4a-2z2 + 4a-2z4 - 4a-2z6 + a-2z8 + a-2z10 + 3a-1z - 6a-1z3 + 2a-1z5 - 3a-1z7 + 2a-1z9 - 2 + 8z2 - 7z4 - 2z6 + 2z8 + 4az3 - 6az5 + 2az7 - 2a2 + 5a2z2 - 4a2z4 + a2z6

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

{-3, -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 1175. 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                     2   j = 13                   3 1   j = 11                 6 2     j = 9               6 3       j = 7             7 6         j = 5           6 6           j = 3         5 7             j = 1       4 7               j = -1     1 4                 j = -3   1 4                   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, 75]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 75]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[12, 6, 13, 5], X[14, 8, 15, 7], > X[18, 9, 19, 10], X[2, 11, 3, 12], X[6, 14, 7, 13], X[22, 16, 1, 15], > X[20, 18, 21, 17], X[8, 19, 9, 20], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 75]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -7, 4, -10, 5, -2, 6, -3, 7, -4, 8, -11, 9, -5, 10, > -9, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 75]]
Out[5]=	DTCode[4, 10, 12, 14, 18, 2, 6, 22, 20, 8, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 75]][t]
Out[6]=	2 10 19 2 3 -21 + -- - -- + -- + 19 t - 10 t + 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 75]][z]
Out[7]=	2 4 6 1 - 3 z + 2 z + 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 75]}
In[9]:=	{KnotDet[Knot[11, Alternating, 75]], KnotSignature[Knot[11, Alternating, 75]]}
Out[9]=	{83, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 75]][q]
Out[10]=	-3 2 5 2 3 4 5 6 7 8 -8 + q - -- + - + 11 q - 13 q + 13 q - 12 q + 9 q - 5 q + 3 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 75]}
In[12]:=	A2Invariant[Knot[11, Alternating, 75]][q]
Out[12]=	-10 -8 2 -2 2 4 6 8 12 14 16 -1 + q + q + -- - q + q - 3 q + 2 q - q + q - 2 q + 3 q + 4 q 22 24 > q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 75]][a, z]
Out[13]=	2 2 4 4 4 6 6 -4 2 2 2 z 3 z 2 2 4 z 3 z 2 z z z -2 + a + 2 a - 5 z - ---- + ---- + a z - 2 z - -- + ---- + ---- + -- + -- 6 4 6 4 2 4 2 a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 75]][a, z]
Out[14]=	2 2 2 2 -4 2 z 2 z 2 z 3 z 2 3 z 2 z 8 z 4 z -2 + a - 2 a - -- - --- + --- + --- + 8 z + ---- + ---- - ---- - ---- + 7 5 3 a 8 6 4 2 a a a a a a a 3 3 3 3 3 4 4 2 2 2 z 4 z 3 z 13 z 6 z 3 4 7 z 3 z > 5 a z - ---- + ---- + ---- - ----- - ---- + 4 a z - 7 z - ---- + ---- + 9 7 5 3 a 8 6 a a a a a a 4 4 5 5 5 5 5 17 z 4 z 2 4 z 8 z 4 z 21 z 2 z 5 6 > ----- + ---- - 4 a z + -- - ---- + ---- + ----- + ---- - 6 a z - 2 z + 4 2 9 7 5 3 a a a a a a a 6 6 6 6 7 7 7 7 3 z 7 z 11 z 4 z 2 6 4 z 5 z 14 z 3 z 7 > ---- - ---- - ----- - ---- + a z + ---- - ---- - ----- - ---- + 2 a z + 8 6 4 2 7 5 3 a a a a a a a a 8 8 8 9 9 9 10 10 8 4 z 3 z z 3 z 5 z 2 z z 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, 75]], Vassiliev[3][Knot[11, Alternating, 75]]}
Out[15]=	{-3, -1}
In[16]:=	Kh[Knot[11, Alternating, 75]][q, t]
Out[16]=	3 1 1 1 4 1 4 4 q 3 7 q + 5 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 7 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 > 6 q t + 6 q t + 7 q t + 6 q t + 6 q t + 3 q t + 6 q t + 11 5 13 5 13 6 15 6 17 7 > 2 q t + 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a75

K11a74 K11a76