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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-10 + 3q-9 - 6q-8 + 9q-7 - 10q-6 + 13q-5 - 12q-4 + 11q-3 - 7q-2 + 5q-1 - 2 + q

A2 (sl(3)) Invariant:

- q-32 - q-30 + 2q-28 - q-26 + 3q-22 - q-20 + 3q-18 + 3q-16 + 3q-14 + 5q-12 + 2q-10 + 5q-8 + 2q-6 - q-4 + 3q-2 + q4

HOMFLY-PT Polynomial:

1 + z2 + a2z-2 + a2 - a2z4 - 2a4z-2 - 3a4 - 2a4z2 - 2a4z4 + a6z-2 - 2a6z2 - 2a6z4 + 2a8 + 3a8z2 - a10

Kauffman Polynomial:

1 - 2z2 + z4 - 2az3 + 2az5 + a2z-2 - 3a2 + 3a2z2 - 3a2z4 + 3a2z6 - 2a3z-1 + 6a3z - 5a3z3 + 3a3z7 + 2a4z-2 - 8a4 + 13a4z2 - 9a4z4 + 3a4z8 - 2a5z-1 + 8a5z - 10a5z3 + 6a5z5 - 5a5z7 + 3a5z9 + a6z-2 - 3a6 - 3a6z2 + 19a6z4 - 21a6z6 + 5a6z8 + a6z10 - a7z3 + 10a7z5 - 17a7z7 + 6a7z9 + 4a8 - 18a8z2 + 38a8z4 - 30a8z6 + 5a8z8 + a8z10 - 4a9z + 11a9z3 - 2a9z5 - 8a9z7 + 3a9z9 + 2a10 - 7a10z2 + 14a10z4 - 12a10z6 + 3a10z8 - 2a11z + 5a11z3 - 4a11z5 + a11z7

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 3                       1 j = 1                     1   j = -1                   4 1   j = -3                 5 3     j = -5               6 2       j = -7             6 5         j = -9           7 6           j = -11         5 8             j = -13       4 5               j = -15     2 5                 j = -17   1 4                   j = -19   2                     j = -21 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, 437]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 437]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[18, 11, 19, 12], X[14, 8, 15, 7], > X[8, 14, 9, 13], X[22, 15, 13, 16], X[20, 17, 21, 18], X[16, 21, 17, 22], > X[12, 19, 5, 20], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 4, -5, 11, -2, 3, -9}, > {5, -4, 6, -8, 7, -3, 9, -7, 8, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-10 3 6 9 10 13 12 11 7 5 -2 - q + -- - -- + -- - -- + -- - -- + -- - -- + - + q 9 8 7 6 5 4 3 2 q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-32 -30 2 -26 3 -20 3 3 3 5 2 5 -q - q + --- - q + --- - q + --- + --- + --- + --- + --- + -- + 28 22 18 16 14 12 10 8 q q q q q q q q 2 -4 3 4 > -- - q + -- + q 6 2 q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 437]][a, z]
Out[8]=	2 4 6 2 4 8 10 a 2 a a 2 4 2 6 2 1 + a - 3 a + 2 a - a + -- - ---- + -- + z - 2 a z - 2 a z + 2 2 2 z z z 8 2 2 4 4 4 6 4 > 3 a z - a z - 2 a z - 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 437]][a, z]
Out[9]=	2 4 6 3 5 2 4 6 8 10 a 2 a a 2 a 2 a 3 1 - 3 a - 8 a - 3 a + 4 a + 2 a + -- + ---- + -- - ---- - ---- + 6 a z + 2 2 2 z z z z z 5 9 11 2 2 2 4 2 6 2 > 8 a z - 4 a z - 2 a z - 2 z + 3 a z + 13 a z - 3 a z - 8 2 10 2 3 3 3 5 3 7 3 9 3 > 18 a z - 7 a z - 2 a z - 5 a z - 10 a z - a z + 11 a z + 11 3 4 2 4 4 4 6 4 8 4 10 4 > 5 a z + z - 3 a z - 9 a z + 19 a z + 38 a z + 14 a z + 5 5 5 7 5 9 5 11 5 2 6 6 6 > 2 a z + 6 a z + 10 a z - 2 a z - 4 a z + 3 a z - 21 a z - 8 6 10 6 3 7 5 7 7 7 9 7 11 7 > 30 a z - 12 a z + 3 a z - 5 a z - 17 a z - 8 a z + a z + 4 8 6 8 8 8 10 8 5 9 7 9 9 9 > 3 a z + 5 a z + 5 a z + 3 a z + 3 a z + 6 a z + 3 a z + 6 10 8 10 > a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 4 1 2 1 4 2 5 4 -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 3 q 21 9 19 8 17 8 17 7 15 7 15 6 13 6 q q t q t q t q t q t q t q t 5 5 8 7 6 6 5 6 2 > ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- + 13 5 11 5 11 4 9 4 9 3 7 3 7 2 5 2 5 q t q t q t q t q t q t q t q t q t 5 t 3 2 > ---- + - + q t + q t 3 q q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a437

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