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DrawMorseLink

PD Presentation:

X6172 X12,3,13,4 X14,8,15,7 X18,10,19,9 X22,19,5,20 X20,15,21,16 X16,21,17,22 X8,18,9,17 X10,14,11,13 X2536 X4,11,1,12

Gauss Code:

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

Jones Polynomial:

q-15/2 - 3q-13/2 + 7q-11/2 - 13q-9/2 + 17q-7/2 - 22q-5/2 + 21q-3/2 - 19q-1/2 + 15q1/2 - 9q3/2 + 4q5/2 - q7/2

A2 (sl(3)) Invariant:

- q-24 - q-22 + q-20 - 2q-18 + q-16 + 5q-14 - q-12 + 6q-10 + 2q-8 + q-6 + 3q-4 - 5q-2 + 3 - 4q2 - q4 + 3q6 - 2q8 + q10

HOMFLY-PT Polynomial:

- 2a-1z - 2a-1z3 - a-1z5 + az-1 + 6az + 8az3 + 4az5 + az7 - 4a3z-1 - 11a3z - 9a3z3 - 3a3z5 + 4a5z-1 + 6a5z + 3a5z3 - a7z-1 - a7z

Kauffman Polynomial:

a-3z3 - a-3z5 - 2a-2z2 + 5a-2z4 - 4a-2z6 + 2a-1z - 7a-1z3 + 12a-1z5 - 8a-1z7 + 1 - 3z4 + 11z6 - 9z8 - az-1 + 5az - 19az3 + 26az5 - 7az7 - 5az9 + 4a2 - 3a2z2 - 15a2z4 + 33a2z6 - 17a2z8 - a2z10 - 4a3z-1 + 13a3z - 30a3z3 + 30a3z5 - a3z7 - 8a3z9 + 7a4 - 12a4z2 - 5a4z4 + 24a4z6 - 12a4z8 - a4z10 - 4a5z-1 + 14a5z - 27a5z3 + 25a5z5 - 5a5z7 - 3a5z9 + 4a6 - 10a6z2 + 5a6z4 + 5a6z6 - 4a6z8 - a7z-1 + 4a7z - 8a7z3 + 8a7z5 - 3a7z7 + a8 - 3a8z2 + 3a8z4 - a8z6

Khovanov Homology:

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 j = 8                       1 j = 6                     3   j = 4                   6 1   j = 2                 9 3     j = 0               10 6       j = -2             12 10         j = -4           10 9           j = -6         7 12             j = -8       6 10               j = -10     2 8                 j = -12   1 5                   j = -14   2                     j = -16 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, 58]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 58]]
Out[4]=	PD[X[6, 1, 7, 2], X[12, 3, 13, 4], X[14, 8, 15, 7], X[18, 10, 19, 9], > X[22, 19, 5, 20], X[20, 15, 21, 16], X[16, 21, 17, 22], X[8, 18, 9, 17], > X[10, 14, 11, 13], X[2, 5, 3, 6], X[4, 11, 1, 12]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -8, 4, -9, 11, -2, 9, -3, 6, -7, 8, -4, > 5, -6, 7, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-(15/2) 3 7 13 17 22 21 19 q - ----- + ----- - ---- + ---- - ---- + ---- - ------- + 15 Sqrt[q] - 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q 3/2 5/2 7/2 > 9 q + 4 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 -20 2 -16 5 -12 6 2 -6 3 5 3 - q - q + q - --- + q + --- - q + --- + -- + q + -- - -- - 18 14 10 8 4 2 q q q q q q 2 4 6 8 10 > 4 q - q + 3 q - 2 q + q
In[8]:=	HOMFLYPT[Link[11, Alternating, 58]][a, z]
Out[8]=	3 5 7 3 a 4 a 4 a a 2 z 3 5 7 2 z 3 - - ---- + ---- - -- - --- + 6 a z - 11 a z + 6 a z - a z - ---- + 8 a z - z z z z a a 5 3 3 5 3 z 5 3 5 7 > 9 a z + 3 a z - -- + 4 a z - 3 a z + a z a
In[9]:=	Kauffman[Link[11, Alternating, 58]][a, z]
Out[9]=	3 5 7 2 4 6 8 a 4 a 4 a a 2 z 3 1 + 4 a + 7 a + 4 a + a - - - ---- - ---- - -- + --- + 5 a z + 13 a z + z z z z a 2 3 5 7 2 z 2 2 4 2 6 2 8 2 z > 14 a z + 4 a z - ---- - 3 a z - 12 a z - 10 a z - 3 a z + -- - 2 3 a a 3 4 7 z 3 3 3 5 3 7 3 4 5 z 2 4 > ---- - 19 a z - 30 a z - 27 a z - 8 a z - 3 z + ---- - 15 a z - a 2 a 5 5 4 4 6 4 8 4 z 12 z 5 3 5 5 5 > 5 a z + 5 a z + 3 a z - -- + ----- + 26 a z + 30 a z + 25 a z + 3 a a 6 7 7 5 6 4 z 2 6 4 6 6 6 8 6 8 z > 8 a z + 11 z - ---- + 33 a z + 24 a z + 5 a z - a z - ---- - 2 a a 7 3 7 5 7 7 7 8 2 8 4 8 6 8 > 7 a z - a z - 5 a z - 3 a z - 9 z - 17 a z - 12 a z - 4 a z - 9 3 9 5 9 2 10 4 10 > 5 a z - 8 a z - 3 a z - a z - a z
In[10]:=	Kh[L][q, t]
Out[10]=	10 1 2 1 5 2 8 6 10 10 + -- + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 2 16 7 14 6 12 6 12 5 10 5 10 4 8 4 8 3 q q t q t q t q t q t q t q t q t 7 12 10 9 12 2 2 2 4 2 > ----- + ----- + ----- + ---- + ---- + 6 t + 9 q t + 3 q t + 6 q t + 6 3 6 2 4 2 4 2 q t q t q t q t q t 4 3 6 3 8 4 > q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a58

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