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

K11a44 K11a46

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X14,6,15,5 X2837 X20,9,21,10 X18,11,19,12 X6,14,7,13 X22,16,1,15 X12,17,13,18 X10,19,11,20 X16,22,17,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 14 2 20 18 6 22 12 10 16

Alexander Polynomial:

- 6t-2 + 22t-1 - 31 + 22t - 6t2

Conway Polynomial:

1 - 2z2 - 6z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{87, 2}

Jones Polynomial:

q-3 - 2q-2 + 5q-1 - 8 + 11q - 14q2 + 14q3 - 12q4 + 10q5 - 6q6 + 3q7 - q8

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

{K11a118, ...}

A2 (sl(3)) Invariant:

q-10 + 3q-4 - q-2 + 2q2 - 4q4 - 2q8 + 2q12 - q14 + 4q16 - q20 + 2q22 - q24 - q26

HOMFLY-PT Polynomial:

- a-8 + 2a-6 + 3a-6z2 + a-4 - a-4z2 - 2a-4z4 - 3a-2 - 5a-2z2 - 3a-2z4 + 1 - z4 + a2 + a2z2

Kauffman Polynomial:

a-9z - 2a-9z3 + a-9z5 - a-8 + 3a-8z2 - 6a-8z4 + 3a-8z6 + a-7z - 2a-7z3 - 5a-7z5 + 4a-7z7 - 2a-6 + 4a-6z2 - 3a-6z4 - 4a-6z6 + 4a-6z8 + a-5z + a-5z5 - 3a-5z7 + 3a-5z9 + a-4 - 13a-4z2 + 28a-4z4 - 19a-4z6 + 5a-4z8 + a-4z10 + 5a-3z - 14a-3z3 + 25a-3z5 - 18a-3z7 + 6a-3z9 + 3a-2 - 23a-2z2 + 37a-2z4 - 22a-2z6 + 4a-2z8 + a-2z10 + 4a-1z - 11a-1z3 + 12a-1z5 - 9a-1z7 + 3a-1z9 + 1 - 5z2 + 8z4 - 9z6 + 3z8 + 3az3 - 6az5 + 2az7 - a2 + 4a2z2 - 4a2z4 + a2z6

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

{-2, 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 1145. 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                   4 1   j = 11                 6 2     j = 9               6 4       j = 7             8 6         j = 5           6 6           j = 3         5 8             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, 45]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 45]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[14, 6, 15, 5], X[2, 8, 3, 7], > X[20, 9, 21, 10], X[18, 11, 19, 12], X[6, 14, 7, 13], X[22, 16, 1, 15], > X[12, 17, 13, 18], X[10, 19, 11, 20], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 45]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -7, 4, -2, 5, -10, 6, -9, 7, -3, 8, -11, 9, -6, 10, > -5, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 45]]
Out[5]=	DTCode[4, 8, 14, 2, 20, 18, 6, 22, 12, 10, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 45]][t]
Out[6]=	6 22 2 -31 - -- + -- + 22 t - 6 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 45]][z]
Out[7]=	2 4 1 - 2 z - 6 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 45]}
In[9]:=	{KnotDet[Knot[11, Alternating, 45]], KnotSignature[Knot[11, Alternating, 45]]}
Out[9]=	{87, 2}
In[10]:=	J=Jones[Knot[11, Alternating, 45]][q]
Out[10]=	-3 2 5 2 3 4 5 6 7 8 -8 + q - -- + - + 11 q - 14 q + 14 q - 12 q + 10 q - 6 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, 45], Knot[11, Alternating, 118]}
In[12]:=	A2Invariant[Knot[11, Alternating, 45]][q]
Out[12]=	-10 3 -2 2 4 8 12 14 16 20 22 q + -- - q + 2 q - 4 q - 2 q + 2 q - q + 4 q - q + 2 q - 4 q 24 26 > q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 45]][a, z]
Out[13]=	2 2 2 4 4 -8 2 -4 3 2 3 z z 5 z 2 2 4 2 z 3 z 1 - a + -- + a - -- + a + ---- - -- - ---- + a z - z - ---- - ---- 6 2 6 4 2 4 2 a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 45]][a, z]
Out[14]=	2 2 -8 2 -4 3 2 z z z 5 z 4 z 2 3 z 4 z 1 - a - -- + a + -- - a + -- + -- + -- + --- + --- - 5 z + ---- + ---- - 6 2 9 7 5 3 a 8 6 a a a a a a a a 2 2 3 3 3 3 13 z 23 z 2 2 2 z 2 z 14 z 11 z 3 4 > ----- - ----- + 4 a z - ---- - ---- - ----- - ----- + 3 a z + 8 z - 4 2 9 7 3 a a a a a a 4 4 4 4 5 5 5 5 5 6 z 3 z 28 z 37 z 2 4 z 5 z z 25 z 12 z > ---- - ---- + ----- + ----- - 4 a z + -- - ---- + -- + ----- + ----- - 8 6 4 2 9 7 5 3 a a a a a a a a a 6 6 6 6 7 7 7 5 6 3 z 4 z 19 z 22 z 2 6 4 z 3 z 18 z > 6 a z - 9 z + ---- - ---- - ----- - ----- + a z + ---- - ---- - ----- - 8 6 4 2 7 5 3 a a a a a a a 7 8 8 8 9 9 9 10 10 9 z 7 8 4 z 5 z 4 z 3 z 6 z 3 z z z > ---- + 2 a z + 3 z + ---- + ---- + ---- + ---- + ---- + ---- + --- + --- a 6 4 2 5 3 a 4 2 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 45]], Vassiliev[3][Knot[11, Alternating, 45]]}
Out[15]=	{-2, 1}
In[16]:=	Kh[Knot[11, Alternating, 45]][q, t]
Out[16]=	3 1 1 1 4 1 4 4 q 3 7 q + 5 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 8 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 + 8 q t + 6 q t + 6 q t + 4 q t + 6 q t + 11 5 13 5 13 6 15 6 17 7 > 2 q t + 4 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a45

K11a44 K11a46