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DrawMorseLink

PD Presentation:

X8192 X18,9,19,10 X6718 X20,13,21,14 X10,4,11,3 X16,6,17,5 X4,12,5,11 X22,15,7,16 X12,19,13,20 X14,21,15,22 X2,18,3,17

Gauss Code:

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

Jones Polynomial:

q-17/2 - 3q-15/2 + 6q-13/2 - 9q-11/2 + 13q-9/2 - 14q-7/2 + 13q-5/2 - 13q-3/2 + 8q-1/2 - 6q1/2 + 3q3/2 - q5/2

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

- 3az-1 - 6az - 8az3 - 5az5 - az7 + 5a3z-1 + 12a3z + 21a3z3 + 18a3z5 + 7a3z7 + a3z9 - 2a5z-1 - 5a5z - 8a5z3 - 5a5z5 - a5z7

Kauffman Polynomial:

a-1z - 4a-1z3 + 4a-1z5 - a-1z7 + 2z2 - 12z4 + 12z6 - 3z8 + 3az-1 - 8az + 14az3 - 20az5 + 16az7 - 4az9 - 5a2 + 12a2z2 - 19a2z4 + 12a2z6 + 2a2z8 - 2a2z10 + 5a3z-1 - 15a3z + 38a3z3 - 50a3z5 + 35a3z7 - 9a3z9 - 5a4 + 12a4z2 - 19a4z4 + 15a4z6 - a4z8 - 2a4z10 + 2a5z-1 - 6a5z + 13a5z3 - 15a5z5 + 12a5z7 - 5a5z9 - 2a6z2 - 5a6z4 + 10a6z6 - 6a6z8 - 4a7z3 + 8a7z5 - 6a7z7 + a8 - 3a8z2 + 6a8z4 - 5a8z6 + 3a9z3 - 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                   4 1   j = 0                 5 3     j = -2               8 3       j = -4             6 6         j = -6           8 7           j = -8         5 6             j = -10       4 8               j = -12     2 5                 j = -14   1 4                   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, 208]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 208]]
Out[4]=	PD[X[8, 1, 9, 2], X[18, 9, 19, 10], X[6, 7, 1, 8], X[20, 13, 21, 14], > X[10, 4, 11, 3], X[16, 6, 17, 5], X[4, 12, 5, 11], X[22, 15, 7, 16], > X[12, 19, 13, 20], X[14, 21, 15, 22], X[2, 18, 3, 17]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -11, 5, -7, 6, -3}, > {3, -1, 2, -5, 7, -9, 4, -10, 8, -6, 11, -2, 9, -4, 10, -8}]
In[6]:=	Jones[L][q]
Out[6]=	-(17/2) 3 6 9 13 14 13 13 8 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 > 6 Sqrt[q] + 3 q - q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -22 2 -18 2 3 3 6 2 5 3 2 4 6 -q + q - --- + q - --- - --- - --- + -- + -- + -- + -- + 2 q - q + q 20 16 14 10 8 6 4 2 q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 208]][a, z]
Out[8]=	3 5 -3 a 5 a 2 a 3 5 3 3 3 5 3 ---- + ---- - ---- - 6 a z + 12 a z - 5 a z - 8 a z + 21 a z - 8 a z - z z z 5 3 5 5 5 7 3 7 5 7 3 9 > 5 a z + 18 a z - 5 a z - a z + 7 a z - a z + a z
In[9]:=	Kauffman[Link[11, Alternating, 208]][a, z]
Out[9]=	3 5 2 4 8 3 a 5 a 2 a z 3 5 2 -5 a - 5 a + a + --- + ---- + ---- + - - 8 a z - 15 a z - 6 a z + 2 z + z z z a 3 2 2 4 2 6 2 8 2 10 2 4 z 3 > 12 a z + 12 a z - 2 a z - 3 a z + a z - ---- + 14 a z + a 3 3 5 3 7 3 9 3 4 2 4 4 4 > 38 a z + 13 a z - 4 a z + 3 a z - 12 z - 19 a z - 19 a z - 5 6 4 8 4 10 4 4 z 5 3 5 5 5 > 5 a z + 6 a z - a z + ---- - 20 a z - 50 a z - 15 a z + a 7 7 5 9 5 6 2 6 4 6 6 6 8 6 z > 8 a z - 3 a z + 12 z + 12 a z + 15 a z + 10 a z - 5 a z - -- + a 7 3 7 5 7 7 7 8 2 8 4 8 > 16 a z + 35 a z + 12 a z - 6 a z - 3 z + 2 a z - a z - 6 8 9 3 9 5 9 2 10 4 10 > 6 a z - 4 a z - 9 a z - 5 a z - 2 a z - 2 a z
In[10]:=	Kh[L][q, t]
Out[10]=	6 8 1 2 1 4 2 5 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 8 5 6 8 7 6 3 t 2 2 2 > ------ + ----- + ----- + ----- + ---- + ---- + 5 t + --- + 3 t + 4 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 L11a208

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