© | Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots:

K11n58 K11n60

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X5,15,6,14 X2837 X16,9,17,10 X18,12,19,11 X20,14,21,13 X15,7,16,6 X22,17,1,18 X12,20,13,19 X10,22,11,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 -14 2 16 18 20 -6 22 12 10

Alexander Polynomial:

- t-3 + 6t-2 - 12t-1 + 15 - 12t + 6t2 - t3

Conway Polynomial:

1 + 3z2 - z6

Other knots with the same Alexander/Conway Polynomial:

{K11n8, ...}

Determinant and Signature:

{53, 4}

Jones Polynomial:

2q2 - 3q3 + 6q4 - 8q5 + 9q6 - 9q7 + 7q8 - 5q9 + 3q10 - q11

Other knots (up to mirrors) with the same Jones Polynomial:

{...}

A2 (sl(3)) Invariant:

2q6 + 3q10 + q12 - q14 + q16 - 3q18 - q22 + 2q26 - q28 + q30 - q34

HOMFLY-PT Polynomial:

- a-10 - a-10z2 + 3a-8 + 5a-8z2 + 2a-8z4 - 5a-6 - 7a-6z2 - 4a-6z4 - a-6z6 + 4a-4 + 6a-4z2 + 2a-4z4

Kauffman Polynomial:

- 2a-13z3 + a-13z5 + 2a-12z2 - 7a-12z4 + 3a-12z6 - 3a-11z + 7a-11z3 - 10a-11z5 + 4a-11z7 + a-10 - 2a-10z2 + 4a-10z4 - 6a-10z6 + 3a-10z8 - 6a-9z + 19a-9z3 - 14a-9z5 + 3a-9z7 + a-9z9 + 3a-8 - 12a-8z2 + 24a-8z4 - 16a-8z6 + 5a-8z8 - 6a-7z + 11a-7z3 - 4a-7z5 + a-7z9 + 5a-6 - 16a-6z2 + 16a-6z4 - 7a-6z6 + 2a-6z8 - 3a-5z + a-5z3 - a-5z5 + a-5z7 + 4a-4 - 8a-4z2 + 3a-4z4

V2 and V3, the type 2 and 3 Vassiliev invariants:

{3, 6}

Khovanov Homology: (The squares with yellow highlighting are those on the "critical diagonals", where j-2r=s+1 or j-2r=s+1, where s=4 is the signature of 1159. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 r = 8 r = 9 j = 23                   1 j = 21                 2   j = 19               3 1   j = 17             4 2     j = 15           5 3       j = 13         4 4         j = 11       4 5           j = 9     2 4             j = 7   1 4               j = 5 1 2                 j = 3 2

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]:=	Show[DrawMorseLink[Knot[11, NonAlternating, 59]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 59]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 15, 6, 14], X[2, 8, 3, 7], > X[16, 9, 17, 10], X[18, 12, 19, 11], X[20, 14, 21, 13], X[15, 7, 16, 6], > X[22, 17, 1, 18], X[12, 20, 13, 19], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 59]]
Out[4]=	GaussCode[1, -4, 2, -1, -3, 8, 4, -2, 5, -11, 6, -10, 7, 3, -8, -5, 9, -6, 10, > -7, 11, -9]
In[5]:=	DTCode[Knot[11, NonAlternating, 59]]
Out[5]=	DTCode[4, 8, -14, 2, 16, 18, 20, -6, 22, 12, 10]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 59]][t]
Out[6]=	-3 6 12 2 3 15 - t + -- - -- - 12 t + 6 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 59]][z]
Out[7]=	2 6 1 + 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 8], Knot[11, NonAlternating, 59]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 59]], KnotSignature[Knot[11, NonAlternating, 59]]}
Out[9]=	{53, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 59]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 2 q - 3 q + 6 q - 8 q + 9 q - 9 q + 7 q - 5 q + 3 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 59]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 59]][q]
Out[12]=	6 10 12 14 16 18 22 26 28 30 34 2 q + 3 q + q - q + q - 3 q - q + 2 q - q + q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 59]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 -10 3 5 4 z 5 z 7 z 6 z 2 z 4 z 2 z z -a + -- - -- + -- - --- + ---- - ---- + ---- + ---- - ---- + ---- - -- 8 6 4 10 8 6 4 8 6 4 6 a a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 59]][a, z]
Out[14]=	2 2 2 2 -10 3 5 4 3 z 6 z 6 z 3 z 2 z 2 z 12 z 16 z a + -- + -- + -- - --- - --- - --- - --- + ---- - ---- - ----- - ----- - 8 6 4 11 9 7 5 12 10 8 6 a a a a a a a a a a a 2 3 3 3 3 3 4 4 4 4 8 z 2 z 7 z 19 z 11 z z 7 z 4 z 24 z 16 z > ---- - ---- + ---- + ----- + ----- + -- - ---- + ---- + ----- + ----- + 4 13 11 9 7 5 12 10 8 6 a a a a a a a a a a 4 5 5 5 5 5 6 6 6 6 3 z z 10 z 14 z 4 z z 3 z 6 z 16 z 7 z > ---- + --- - ----- - ----- - ---- - -- + ---- - ---- - ----- - ---- + 4 13 11 9 7 5 12 10 8 6 a a a a a a a a a a 7 7 7 8 8 8 9 9 4 z 3 z z 3 z 5 z 2 z z z > ---- + ---- + -- + ---- + ---- + ---- + -- + -- 11 9 5 10 8 6 9 7 a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 59]], Vassiliev[3][Knot[11, NonAlternating, 59]]}
Out[15]=	{3, 6}
In[16]:=	Kh[Knot[11, NonAlternating, 59]][q, t]
Out[16]=	3 5 5 7 7 2 9 2 9 3 11 3 11 4 2 q + q + 2 q t + q t + 4 q t + 2 q t + 4 q t + 4 q t + 5 q t + 13 4 13 5 15 5 15 6 17 6 17 7 > 4 q t + 4 q t + 5 q t + 3 q t + 4 q t + 2 q t + 19 7 19 8 21 8 23 9 > 3 q t + q t + 2 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n59

K11n58 K11n60