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DrawMorseLink

PD Presentation:

X6172 X16,7,17,8 X18,9,19,10 X20,11,21,12 X8,17,9,18 X10,19,11,20 X4,21,1,22 X14,6,15,5 X12,4,13,3 X22,14,5,13 X2,16,3,15

Gauss Code:

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

Jones Polynomial:

q-17/2 - 3q-15/2 + 5q-13/2 - 7q-11/2 + 10q-9/2 - 11q-7/2 + 10q-5/2 - 10q-3/2 + 6q-1/2 - 5q1/2 + 3q3/2 - q5/2

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

- 2az-1 - 4az - 7az3 - 5az5 - az7 + 3a3z-1 + 7a3z + 17a3z3 + 17a3z5 + 7a3z7 + a3z9 - a5z-1 - 3a5z - 7a5z3 - 5a5z5 - a5z7

Kauffman Polynomial:

- 3a-1z3 + 4a-1z5 - a-1z7 + 2z2 - 13z4 + 13z6 - 3z8 + 2az-1 - 7az + 19az3 - 28az5 + 19az7 - 4az9 - 3a2 + 6a2z2 - 11a2z4 + 4a2z6 + 5a2z8 - 2a2z10 + 3a3z-1 - 13a3z + 46a3z3 - 63a3z5 + 38a3z7 - 8a3z9 - 3a4 + 8a4z2 - 11a4z4 + 3a4z6 + 4a4z8 - 2a4z10 + a5z-1 - 6a5z + 18a5z3 - 23a5z5 + 14a5z7 - 4a5z9 - a6 + 2a6z2 - 7a6z4 + 8a6z6 - 4a6z8 - 2a7z3 + 5a7z5 - 4a7z7 - a8z2 + 5a8z4 - 4a8z6 + 4a9z3 - 3a9z5 + a10z2 - a10z4

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 = 6                       1 j = 4                     2   j = 2                   3 1   j = 0                 3 2     j = -2               7 3       j = -4             5 5         j = -6           6 5           j = -8         4 5             j = -10       3 6               j = -12     2 4                 j = -14   1 3                   j = -16   2                     j = -18 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, 100]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 100]]
Out[4]=	PD[X[6, 1, 7, 2], X[16, 7, 17, 8], X[18, 9, 19, 10], X[20, 11, 21, 12], > X[8, 17, 9, 18], X[10, 19, 11, 20], X[4, 21, 1, 22], X[14, 6, 15, 5], > X[12, 4, 13, 3], X[22, 14, 5, 13], X[2, 16, 3, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 9, -7}, {8, -1, 2, -5, 3, -6, 4, -9, 10, -8, 11, -2, 5, -3, > 6, -4, 7, -10}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 3 5 7 10 11 10 10 6 q - ----- + ----- - ----- + ---- - ---- + ---- - ---- + ------- - 15/2 13/2 11/2 9/2 7/2 5/2 3/2 Sqrt[q] q q q q q q q 3/2 5/2 > 5 Sqrt[q] + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 -20 -18 -16 2 2 5 2 4 2 2 4 6 -q + q - q + q - q - --- - --- + -- + -- + -- + -- + q - q + q 14 10 8 6 4 2 q q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 100]][a, z]
Out[8]=	3 5 -2 a 3 a a 3 5 3 3 3 5 3 ---- + ---- - -- - 4 a z + 7 a z - 3 a z - 7 a z + 17 a z - 7 a z - z z z 5 3 5 5 5 7 3 7 5 7 3 9 > 5 a z + 17 a z - 5 a z - a z + 7 a z - a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 100]][a, z]
Out[9]=	3 5 2 4 6 2 a 3 a a 3 5 2 -3 a - 3 a - a + --- + ---- + -- - 7 a z - 13 a z - 6 a z + 2 z + z z z 3 2 2 4 2 6 2 8 2 10 2 3 z 3 3 3 > 6 a z + 8 a z + 2 a z - a z + a z - ---- + 19 a z + 46 a z + a 5 3 7 3 9 3 4 2 4 4 4 6 4 > 18 a z - 2 a z + 4 a z - 13 z - 11 a z - 11 a z - 7 a z + 5 8 4 10 4 4 z 5 3 5 5 5 7 5 > 5 a z - a z + ---- - 28 a z - 63 a z - 23 a z + 5 a z - a 7 9 5 6 2 6 4 6 6 6 8 6 z 7 > 3 a z + 13 z + 4 a z + 3 a z + 8 a z - 4 a z - -- + 19 a z + a 3 7 5 7 7 7 8 2 8 4 8 6 8 > 38 a z + 14 a z - 4 a z - 3 z + 5 a z + 4 a z - 4 a z - 9 3 9 5 9 2 10 4 10 > 4 a z - 8 a z - 4 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	5 7 1 2 1 3 2 4 3 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 4 2 18 7 16 6 14 6 14 5 12 5 12 4 10 4 q q q t q t q t q t q t q t q t 6 4 5 6 5 5 3 t 2 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + 3 t + --- + 2 t + 3 q t + 10 3 8 3 8 2 6 2 6 4 2 q t q t q t q t q t q t q 2 3 4 3 6 4 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a100

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