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DrawMorseLink

PD Presentation:

X6172 X10,3,11,4 X11,16,12,17 X21,18,22,19 X13,20,14,21 X19,12,20,13 X17,22,18,9 X8,16,5,15 X14,8,15,7 X2536 X4,9,1,10

Gauss Code:

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

Jones Polynomial:

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

A2 (sl(3)) Invariant:

- q-32 - 2q-30 - q-28 - 4q-26 - 3q-24 + 5q-18 + 5q-16 + 6q-14 + 6q-12 + 4q-10 + 6q-8 + 3q-6 + q-4 + 2q-2

HOMFLY-PT Polynomial:

a2z-2 + 3a2 + 2a2z2 - a4z-2 - 2a4 - a4z2 - a4z4 - 2a6z-2 - 5a6 - 5a6z2 - 2a6z4 + 3a8z-2 + 5a8 + 3a8z2 - a10z-2 - a10

Kauffman Polynomial:

a2z-2 - 4a2 + 3a2z2 - 2a3z-1 + 4a3z - a3z3 + a3z5 + a4z-2 - 3a4 + 4a4z2 - 3a4z4 + 2a4z6 + 2a5z-1 - 10a5z + 17a5z3 - 13a5z5 + 4a5z7 - 2a6z-2 + 4a6 - 4a6z2 + 11a6z4 - 13a6z6 + 4a6z8 + 10a7z-1 - 34a7z + 49a7z3 - 33a7z5 + 5a7z7 + a7z9 - 3a8z-2 + 5a8 - 7a8z2 + 22a8z4 - 23a8z6 + 6a8z8 + 8a9z-1 - 27a9z + 40a9z3 - 24a9z5 + 2a9z7 + a9z9 - a10z-2 + a10 - 2a10z2 + 8a10z4 - 8a10z6 + 2a10z8 + 2a11z-1 - 7a11z + 9a11z3 - 5a11z5 + a11z7

Khovanov Homology:

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -1                   2 j = -3                 3 3 j = -5               3   1 j = -7             4 4     j = -9           5 3       j = -11         2 5 1       j = -13       4 4           j = -15     1 2             j = -17   1 4               j = -19   1                 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, NonAlternating, 257]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, NonAlternating, 257]]
Out[4]=	PD[X[6, 1, 7, 2], X[10, 3, 11, 4], X[11, 16, 12, 17], X[21, 18, 22, 19], > X[13, 20, 14, 21], X[19, 12, 20, 13], X[17, 22, 18, 9], X[8, 16, 5, 15], > X[14, 8, 15, 7], X[2, 5, 3, 6], X[4, 9, 1, 10]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 2, -11}, {10, -1, 9, -8}, > {11, -2, -3, 6, -5, -9, 8, 3, -7, 4, -6, 5, -4, 7}]
In[6]:=	Jones[L][q]
Out[6]=	-10 2 5 6 6 8 6 6 2 2 -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 2 -28 4 3 5 5 6 6 4 6 3 -4 -q - --- - q - --- - --- + --- + --- + --- + --- + --- + -- + -- + q + 30 26 24 18 16 14 12 10 8 6 q q q q q q q q q q 2 > -- 2 q
In[8]:=	HOMFLYPT[Link[11, NonAlternating, 257]][a, z]
Out[8]=	2 4 6 8 10 2 4 6 8 10 a a 2 a 3 a a 2 2 3 a - 2 a - 5 a + 5 a - a + -- - -- - ---- + ---- - --- + 2 a z - 2 2 2 2 2 z z z z z 4 2 6 2 8 2 4 4 6 4 > a z - 5 a z + 3 a z - a z - 2 a z
In[9]:=	Kauffman[Link[11, NonAlternating, 257]][a, z]
Out[9]=	2 4 6 8 10 3 5 2 4 6 8 10 a a 2 a 3 a a 2 a 2 a -4 a - 3 a + 4 a + 5 a + a + -- + -- - ---- - ---- - --- - ---- + ---- + 2 2 2 2 2 z z z z z z z 7 9 11 10 a 8 a 2 a 3 5 7 9 11 > ----- + ---- + ----- + 4 a z - 10 a z - 34 a z - 27 a z - 7 a z + z z z 2 2 4 2 6 2 8 2 10 2 3 3 5 3 > 3 a z + 4 a z - 4 a z - 7 a z - 2 a z - a z + 17 a z + 7 3 9 3 11 3 4 4 6 4 8 4 10 4 > 49 a z + 40 a z + 9 a z - 3 a z + 11 a z + 22 a z + 8 a z + 3 5 5 5 7 5 9 5 11 5 4 6 6 6 > a z - 13 a z - 33 a z - 24 a z - 5 a z + 2 a z - 13 a z - 8 6 10 6 5 7 7 7 9 7 11 7 6 8 > 23 a z - 8 a z + 4 a z + 5 a z + 2 a z + a z + 4 a z + 8 8 10 8 7 9 9 9 > 6 a z + 2 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	-5 3 2 1 1 1 4 1 2 4 q + -- + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 4 2 5 5 1 3 4 4 3 > ------ + ------ + ------ + ----- + ------ + ----- + ----- + ----- + ----- + 13 5 11 5 11 4 9 4 11 3 9 3 7 3 7 2 5 2 q t q t q t q t q t q t q t q t q t 3 > ---- 3 q t

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11n257

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