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DrawMorseLink

PD Presentation:

X8192 X16,5,17,6 X22,15,13,16 X14,4,15,3 X4,14,5,13 X12,17,7,18 X10,19,11,20 X18,9,19,10 X20,11,21,12 X2738 X6,21,1,22

Gauss Code:

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

Jones Polynomial:

- q-11 + 4q-10 - 8q-9 + 13q-8 - 17q-7 + 20q-6 - 18q-5 + 17q-4 - 11q-3 + 7q-2 - 3q-1 + 1

A2 (sl(3)) Invariant:

- q-34 + 2q-30 - 2q-28 + 3q-26 + 2q-24 + 6q-20 + 6q-16 + 3q-14 + 2q-12 + 6q-10 - 2q-8 + 2q-6 - q-2 + 1

HOMFLY-PT Polynomial:

2a2z2 + a2z4 + a4z-2 + 5a4 + 4a4z2 - a4z4 - a4z6 - 2a6z-2 - 8a6 - 10a6z2 - 7a6z4 - 2a6z6 + a8z-2 + 4a8 + 7a8z2 + 3a8z4 - a10 - a10z2

Kauffman Polynomial:

2a2z2 - 3a2z4 + a2z6 + 5a3z3 - 8a3z5 + 3a3z7 - a4z-2 + 6a4 - 13a4z2 + 16a4z4 - 14a4z6 + 5a4z8 + 2a5z-1 - 6a5z + 7a5z3 - 3a5z5 - 5a5z7 + 4a5z9 - 2a6z-2 + 12a6 - 37a6z2 + 53a6z4 - 40a6z6 + 12a6z8 + a6z10 + 2a7z-1 - 8a7z + 14a7z3 - 6a7z5 - 9a7z7 + 8a7z9 - a8z-2 + 8a8 - 24a8z2 + 39a8z4 - 36a8z6 + 14a8z8 + a8z10 - 3a9z + 18a9z3 - 23a9z5 + 6a9z7 + 4a9z9 + a10 - a10z4 - 7a10z6 + 7a10z8 - a11z + 5a11z3 - 11a11z5 + 7a11z7 + 2a12z2 - 6a12z4 + 4a12z6 - a13z3 + 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                     2   j = -3                   5 1   j = -5                 7 3     j = -7               10 4       j = -9             9 8         j = -11           11 9           j = -13         7 10             j = -15       6 10               j = -17     3 8                 j = -19   1 5                   j = -21   3                     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, 512]]]
Out[3]=	-Graphics-
In[4]:=	PD[L = Link[11, Alternating, 512]]
Out[4]=	PD[X[8, 1, 9, 2], X[16, 5, 17, 6], X[22, 15, 13, 16], X[14, 4, 15, 3], > X[4, 14, 5, 13], X[12, 17, 7, 18], X[10, 19, 11, 20], X[18, 9, 19, 10], > X[20, 11, 21, 12], X[2, 7, 3, 8], X[6, 21, 1, 22]]
In[5]:=	GaussCode[L]
Out[5]=	GaussCode[{1, -10, 4, -5, 2, -11}, {10, -1, 8, -7, 9, -6}, > {5, -4, 3, -2, 6, -8, 7, -9, 11, -3}]
In[6]:=	Jones[L][q]
Out[6]=	-11 4 8 13 17 20 18 17 11 7 3 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]=	-34 2 2 3 2 6 6 3 2 6 2 2 -2 1 - q + --- - --- + --- + --- + --- + --- + --- + --- + --- - -- + -- - q 30 28 26 24 20 16 14 12 10 8 6 q q q q q q q q q q q
In[8]:=	HOMFLYPT[Link[11, Alternating, 512]][a, z]
Out[8]=	4 6 8 4 6 8 10 a 2 a a 2 2 4 2 6 2 5 a - 8 a + 4 a - a + -- - ---- + -- + 2 a z + 4 a z - 10 a z + 2 2 2 z z z 8 2 10 2 2 4 4 4 6 4 8 4 4 6 6 6 > 7 a z - a z + a z - a z - 7 a z + 3 a z - a z - 2 a z
In[9]:=	Kauffman[Link[11, Alternating, 512]][a, z]
Out[9]=	4 6 8 5 7 4 6 8 10 a 2 a a 2 a 2 a 5 7 6 a + 12 a + 8 a + a - -- - ---- - -- + ---- + ---- - 6 a z - 8 a z - 2 2 2 z z z z z 9 11 2 2 4 2 6 2 8 2 12 2 > 3 a z - a z + 2 a z - 13 a z - 37 a z - 24 a z + 2 a z + 3 3 5 3 7 3 9 3 11 3 13 3 2 4 > 5 a z + 7 a z + 14 a z + 18 a z + 5 a z - a z - 3 a z + 4 4 6 4 8 4 10 4 12 4 3 5 5 5 > 16 a z + 53 a z + 39 a z - a z - 6 a z - 8 a z - 3 a z - 7 5 9 5 11 5 13 5 2 6 4 6 6 6 > 6 a z - 23 a z - 11 a z + a z + a z - 14 a z - 40 a z - 8 6 10 6 12 6 3 7 5 7 7 7 9 7 > 36 a z - 7 a z + 4 a z + 3 a z - 5 a z - 9 a z + 6 a z + 11 7 4 8 6 8 8 8 10 8 5 9 7 9 > 7 a z + 5 a z + 12 a z + 14 a z + 7 a z + 4 a z + 8 a z + 9 9 6 10 8 10 > 4 a z + a z + a z
In[10]:=	Kh[L][q, t]
Out[10]=	3 5 1 3 1 5 3 8 6 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 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 10 7 10 11 9 9 8 10 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 2 t 2 > ---- + -- + --- + q t 5 3 q q t q

Dror Bar-Natan: The Knot Atlas: The Thistlethwaite Link Table: The Link L11a512

L11a511 L11a513