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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-8 + 3q-7 - 6q-6 + 10q-5 - 12q-4 + 15q-3 - 14q-2 + 13q-1 - 8 + 6q - 3q2 + q3

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

z-2 + 4 + 5z2 + 4z4 + z6 - 2a2z-2 - 9a2 - 16a2z2 - 14a2z4 - 6a2z6 - a2z8 + a4z-2 + 7a4 + 13a4z2 + 9a4z4 + 2a4z6 - 2a6 - 3a6z2 - a6z4

Kauffman Polynomial:

a-2z2 - 3a-2z4 + a-2z6 + 4a-1z3 - 9a-1z5 + 3a-1z7 - z-2 + 6 - 14z2 + 21z4 - 18z6 + 5z8 + 2az-1 - 5az + 2az3 + 7az5 - 11az7 + 4az9 - 2a2z-2 + 13a2 - 44a2z2 + 67a2z4 - 41a2z6 + 8a2z8 + a2z10 + 2a3z-1 - 8a3z + 4a3z3 + 19a3z5 - 20a3z7 + 7a3z9 - a4z-2 + 9a4 - 31a4z2 + 47a4z4 - 29a4z6 + 7a4z8 + a4z10 - 3a5z + 6a5z3 - 3a5z5 - 2a5z7 + 3a5z9 + a6z2 - 2a6z4 - 4a6z6 + 4a6z8 + a7z - 2a7z3 - 5a7z5 + 4a7z7 - a8 + 3a8z2 - 6a8z4 + 3a8z6 + a9z - 2a9z3 + a9z5

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 = 7                       1 j = 5                     2   j = 3                   4 1   j = 1                 4 2     j = -1               9 4       j = -3             8 7         j = -5           7 6           j = -7         5 8             j = -9       5 7               j = -11     2 6                 j = -13   1 4                   j = -15   2                     j = -17 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]=	3
In[3]:=	Show[DrawMorseLink[Link[11, Alternating, 446]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 446]]
Out[4]=	PD[X[6, 1, 7, 2], X[14, 4, 15, 3], X[18, 9, 19, 10], X[16, 8, 17, 7], > X[22, 17, 13, 18], X[20, 11, 21, 12], X[8, 14, 9, 13], X[10, 19, 11, 20], > X[12, 21, 5, 22], X[2, 5, 3, 6], X[4, 16, 1, 15]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -7, 3, -8, 6, -9}, > {7, -2, 11, -4, 5, -3, 8, -6, 9, -5}]
In[6]:=	Jones[L][q]
Out[6]=	-8 3 6 10 12 15 14 13 2 3 -8 - q + -- - -- + -- - -- + -- - -- + -- + 6 q - 3 q + q 7 6 5 4 3 2 q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-24 -18 3 4 4 2 6 5 2 4 6 8 2 - q - q + --- + --- + --- + -- + -- + -- + q + 2 q - q + q 16 12 10 8 6 2 q q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 446]][a, z]
Out[8]=	2 4 2 4 6 -2 2 a a 2 2 2 4 2 4 - 9 a + 7 a - 2 a + z - ---- + -- + 5 z - 16 a z + 13 a z - 2 2 z z 6 2 4 2 4 4 4 6 4 6 2 6 4 6 2 8 > 3 a z + 4 z - 14 a z + 9 a z - a z + z - 6 a z + 2 a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 446]][a, z]
Out[9]=	2 4 3 2 4 8 -2 2 a a 2 a 2 a 3 6 + 13 a + 9 a - a - z - ---- - -- + --- + ---- - 5 a z - 8 a z - 2 2 z z z z 2 5 7 9 2 z 2 2 4 2 6 2 8 2 > 3 a z + a z + a z - 14 z + -- - 44 a z - 31 a z + a z + 3 a z + 2 a 3 4 4 z 3 3 3 5 3 7 3 9 3 4 3 z > ---- + 2 a z + 4 a z + 6 a z - 2 a z - 2 a z + 21 z - ---- + a 2 a 5 2 4 4 4 6 4 8 4 9 z 5 3 5 > 67 a z + 47 a z - 2 a z - 6 a z - ---- + 7 a z + 19 a z - a 6 5 5 7 5 9 5 6 z 2 6 4 6 6 6 > 3 a z - 5 a z + a z - 18 z + -- - 41 a z - 29 a z - 4 a z + 2 a 7 8 6 3 z 7 3 7 5 7 7 7 8 2 8 > 3 a z + ---- - 11 a z - 20 a z - 2 a z + 4 a z + 5 z + 8 a z + a 4 8 6 8 9 3 9 5 9 2 10 4 10 > 7 a z + 4 a z + 4 a z + 7 a z + 3 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	7 9 1 2 1 4 2 6 5 7 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ----- + ----- + 3 q 17 7 15 6 13 6 13 5 11 5 11 4 9 4 9 3 q q t q t q t q t q t q t q t q t 5 8 7 6 8 4 t 2 3 2 > ----- + ----- + ----- + ---- + ---- + --- + 4 q t + 2 q t + 4 q t + 7 3 7 2 5 2 5 3 q q t q t q t q t q t 3 3 5 3 7 4 > q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a446

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