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DrawMorseLink

PD Presentation:

X12,1,13,2 X14,3,15,4 X20,10,21,9 X16,6,17,5 X18,8,19,7 X22,15,11,16 X6,18,7,17 X8,20,9,19 X4,22,5,21 X2,11,3,12 X10,13,1,14

Gauss Code:

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

Jones Polynomial:

q-7/2 - 2q-5/2 + 3q-3/2 - 6q-1/2 + 7q1/2 - 9q3/2 + 9q5/2 - 9q7/2 + 7q9/2 - 5q11/2 + 3q13/2 - q15/2

A2 (sl(3)) Invariant:

- q-10 + 2q-2 + 1 + 3q2 + q4 + 2q8 - q10 + 2q12 - q20 + q22

HOMFLY-PT Polynomial:

- a-5z - 3a-5z3 - a-5z5 - a-3z-1 - a-3z + 3a-3z3 + 4a-3z5 + a-3z7 + a-1z-1 + 6a-1z + 8a-1z3 + 5a-1z5 + a-1z7 - 3az - 4az3 - az5

Kauffman Polynomial:

- a-9z3 + a-8z2 - 3a-8z4 - a-7z + 4a-7z3 - 5a-7z5 - a-6z2 + 7a-6z4 - 6a-6z6 - 2a-5z3 + 10a-5z5 - 6a-5z7 + a-4z2 - 4a-4z4 + 11a-4z6 - 5a-4z8 + a-3z-1 - 3a-3z - a-3z3 + 2a-3z5 + 6a-3z7 - 3a-3z9 - a-2 + 7a-2z2 - 23a-2z4 + 19a-2z6 - 2a-2z8 - a-2z10 + a-1z-1 - 9a-1z + 25a-1z3 - 37a-1z5 + 24a-1z7 - 5a-1z9 + 10z2 - 20z4 + 8z6 + 2z8 - z10 - 5az + 19az3 - 24az5 + 12az7 - 2az9 + 6a2z2 - 11a2z4 + 6a2z6 - a2z8

Khovanov Homology:

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 16                       1 j = 14                     2   j = 12                   3 1   j = 10                 4 2     j = 8               5 3       j = 6             5 5         j = 4           4 4           j = 2         4 6             j = 0       2 3               j = -2     1 4                 j = -4   1 2                   j = -6   1                     j = -8 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]:=	Length[Skeleton[L]]
Out[2]=	2
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 358]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 358]]
Out[4]=	PD[X[12, 1, 13, 2], X[14, 3, 15, 4], X[20, 10, 21, 9], X[16, 6, 17, 5], > X[18, 8, 19, 7], X[22, 15, 11, 16], X[6, 18, 7, 17], X[8, 20, 9, 19], > X[4, 22, 5, 21], X[2, 11, 3, 12], X[10, 13, 1, 14]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -9, 4, -7, 5, -8, 3, -11}, > {10, -1, 11, -2, 6, -4, 7, -5, 8, -3, 9, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-(7/2) 2 3 6 3/2 5/2 7/2 q - ---- + ---- - ------- + 7 Sqrt[q] - 9 q + 9 q - 9 q + 5/2 3/2 Sqrt[q] q q 9/2 11/2 13/2 15/2 > 7 q - 5 q + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-10 2 2 4 8 10 12 20 22 1 - q + -- + 3 q + q + 2 q - q + 2 q - q + q 2 q
In[8]:=	HOMFLYPT[Link[11, Alternating, 358]][a, z]
Out[8]=	3 3 3 5 1 1 z z 6 z 3 z 3 z 8 z 3 z -(----) + --- - -- - -- + --- - 3 a z - ---- + ---- + ---- - 4 a z - -- + 3 a z 5 3 a 5 3 a 5 a z a a a a a 5 5 7 7 4 z 5 z 5 z z > ---- + ---- - a z + -- + -- 3 a 3 a a a
In[9]:=	Kauffman[Link[11, Alternating, 358]][a, z]
Out[9]=	2 2 2 2 -2 1 1 z 3 z 9 z 2 z z z 7 z -a + ---- + --- - -- - --- - --- - 5 a z + 10 z + -- - -- + -- + ---- + 3 a z 7 3 a 8 6 4 2 a z a a a a a a 3 3 3 3 3 4 4 2 2 z 4 z 2 z z 25 z 3 4 3 z 7 z > 6 a z - -- + ---- - ---- - -- + ----- + 19 a z - 20 z - ---- + ---- - 9 7 5 3 a 8 6 a a a a a a 4 4 5 5 5 5 4 z 23 z 2 4 5 z 10 z 2 z 37 z 5 6 > ---- - ----- - 11 a z - ---- + ----- + ---- - ----- - 24 a z + 8 z - 4 2 7 5 3 a a a a a a 6 6 6 7 7 7 6 z 11 z 19 z 2 6 6 z 6 z 24 z 7 8 > ---- + ----- + ----- + 6 a z - ---- + ---- + ----- + 12 a z + 2 z - 6 4 2 5 3 a a a a a a 8 8 9 9 10 5 z 2 z 2 8 3 z 5 z 9 10 z > ---- - ---- - a z - ---- - ---- - 2 a z - z - --- 4 2 3 a 2 a a a a
In[10]:=	Kh[L][q, t]
Out[10]=	2 2 4 1 1 1 2 1 2 4 3 4 q 6 q + 4 q + ----- + ----- + ----- + ----- + ----- + -- + ----- + - + ---- + 8 5 6 4 4 4 4 3 2 3 2 2 2 t t q t q t q t q t q t t q t 4 6 6 2 8 2 8 3 10 3 10 4 > 4 q t + 5 q t + 5 q t + 5 q t + 3 q t + 4 q t + 2 q t + 12 4 12 5 14 5 16 6 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a358

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