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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

Jones Polynomial:

- q-11 + 2q-10 - 5q-9 + 9q-8 - 12q-7 + 17q-6 - 15q-5 + 15q-4 - 11q-3 + 8q-2 - 4q-1 + 1

A2 (sl(3)) Invariant:

- q-36 - 3q-34 - 2q-32 - q-30 - 2q-28 + 4q-26 + 6q-24 + 4q-22 + 9q-20 + 4q-18 + 5q-16 + 3q-14 - q-12 + 4q-10 - 3q-8 + q-6 + q-4 - 2q-2 + 1

HOMFLY-PT Polynomial:

a2z2 + a2z4 - a4z2 - 2a4z4 - a4z6 + 2a6z-2 + 7a6 + 9a6z2 + 4a6z4 - 5a8z-2 - 11a8 - 6a8z2 + 4a10z-2 + 4a10 - a12z-2

Kauffman Polynomial:

a2z2 - 2a2z4 + a2z6 + 5a3z3 - 10a3z5 + 4a3z7 - 2a4z2 + 8a4z4 - 15a4z6 + 6a4z8 + 6a5z3 - 8a5z5 - 4a5z7 + 4a5z9 + 2a6z-2 - 7a6 + 6a6z2 + 11a6z4 - 22a6z6 + 8a6z8 + a6z10 - 5a7z-1 + 11a7z - 6a7z3 + 4a7z5 - 9a7z7 + 6a7z9 + 5a8z-2 - 14a8 + 20a8z2 - 8a8z4 - 4a8z6 + 4a8z8 + a8z10 - 9a9z-1 + 21a9z - 16a9z3 + 3a9z5 + a9z7 + 2a9z9 + 4a10z-2 - 10a10 + 14a10z2 - 13a10z4 + 4a10z6 + 2a10z8 - 5a11z-1 + 13a11z - 12a11z3 + 2a11z5 + 2a11z7 + a12z-2 - 2a12 + 3a12z2 - 4a12z4 + 2a12z6 - a13z-1 + 3a13z - 3a13z3 + a13z5

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 = 1                       1 j = -1                     3   j = -3                   5 1   j = -5                 7 4     j = -7               8 4       j = -9             7 7         j = -11           10 8           j = -13         5 10             j = -15       4 7               j = -17     1 5                 j = -19   1 4                   j = -21   1                     j = -23 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, 395]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 395]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[16, 7, 17, 8], X[8, 15, 5, 16], > X[20, 11, 21, 12], X[22, 19, 9, 20], X[18, 14, 19, 13], X[14, 18, 15, 17], > X[12, 21, 13, 22], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 3, -4}, > {11, -2, 5, -9, 7, -8, 4, -3, 8, -7, 6, -5, 9, -6}]
In[6]:=	Jones[L][q]
Out[6]=	-11 2 5 9 12 17 15 15 11 8 4 1 - q + --- - -- + -- - -- + -- - -- + -- - -- + -- - - 10 9 8 7 6 5 4 3 2 q q q q q q q q q q
In[7]:=	A2Invariant[L][q]
Out[7]=	-36 3 2 -30 2 4 6 4 9 4 5 3 1 - q - --- - --- - q - --- + --- + --- + --- + --- + --- + --- + --- - 34 32 28 26 24 22 20 18 16 14 q q q q q q q q q q -12 4 3 -6 -4 2 > q + --- - -- + q + q - -- 10 8 2 q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 395]][a, z]
Out[8]=	6 8 10 12 6 8 10 2 a 5 a 4 a a 2 2 4 2 6 2 7 a - 11 a + 4 a + ---- - ---- + ----- - --- + a z - a z + 9 a z - 2 2 2 2 z z z z 8 2 2 4 4 4 6 4 4 6 > 6 a z + a z - 2 a z + 4 a z - a z
In[9]:=	Kauffman[Link[11, Alternating, 395]][a, z]
Out[9]=	6 8 10 12 7 9 6 8 10 12 2 a 5 a 4 a a 5 a 9 a -7 a - 14 a - 10 a - 2 a + ---- + ---- + ----- + --- - ---- - ---- - 2 2 2 2 z z z z z z 11 13 5 a a 7 9 11 13 2 2 4 2 > ----- - --- + 11 a z + 21 a z + 13 a z + 3 a z + a z - 2 a z + z z 6 2 8 2 10 2 12 2 3 3 5 3 7 3 > 6 a z + 20 a z + 14 a z + 3 a z + 5 a z + 6 a z - 6 a z - 9 3 11 3 13 3 2 4 4 4 6 4 8 4 > 16 a z - 12 a z - 3 a z - 2 a z + 8 a z + 11 a z - 8 a z - 10 4 12 4 3 5 5 5 7 5 9 5 11 5 > 13 a z - 4 a z - 10 a z - 8 a z + 4 a z + 3 a z + 2 a z + 13 5 2 6 4 6 6 6 8 6 10 6 12 6 > a z + a z - 15 a z - 22 a z - 4 a z + 4 a z + 2 a z + 3 7 5 7 7 7 9 7 11 7 4 8 6 8 > 4 a z - 4 a z - 9 a z + a z + 2 a z + 6 a z + 8 a z + 8 8 10 8 5 9 7 9 9 9 6 10 8 10 > 4 a z + 2 a z + 4 a z + 6 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	4 5 1 1 1 4 1 5 4 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 23 9 21 8 19 8 19 7 17 7 17 6 15 6 q q q t q t q t q t q t q t q t 7 5 10 10 8 7 7 8 4 > ------ + ------ + ------ + ------ + ------ + ----- + ----- + ----- + ---- + 15 5 13 5 13 4 11 4 11 3 9 3 9 2 7 2 7 q t q t q t q t q t q t q t q t q t 7 t 3 t 2 > ---- + -- + --- + q t 5 3 q q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a395

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