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

K11n28 K11n30

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X5,13,6,12 X2837 X14,9,15,10 X18,12,19,11 X13,7,14,6 X22,15,1,16 X20,17,21,18 X10,20,11,19 X16,21,17,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- 3t-2 + 13t-1 - 19 + 13t - 3t2

Conway Polynomial:

1 + z2 - 3z4

Other knots with the same Alexander/Conway Polynomial:

{1036, K11a230, ...}

Determinant and Signature:

{51, 2}

Jones Polynomial:

2q-1 - 4 + 6q - 8q2 + 9q3 - 8q4 + 7q5 - 4q6 + 2q7 - q8

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

{...}

A2 (sl(3)) Invariant:

2q-4 + 2q2 - 3q4 - q8 + 2q12 + 3q16 - q20 + q22 - q24 - q26

HOMFLY-PT Polynomial:

- a-8 + a-6 + 2a-6z2 + 2a-4 + a-4z2 - a-4z4 - 3a-2 - 4a-2z2 - 2a-2z4 + 2 + 2z2

Kauffman Polynomial:

2a-9z - 3a-9z3 + a-9z5 - a-8 + 3a-8z2 - 5a-8z4 + 2a-8z6 + 2a-7z - 3a-7z3 - 2a-7z5 + 2a-7z7 - a-6 + 2a-6z4 - 3a-6z6 + 2a-6z8 + a-5z + a-5z9 + 2a-4 - 11a-4z2 + 18a-4z4 - 11a-4z6 + 4a-4z8 + 3a-3z - 5a-3z3 + 4a-3z5 - a-3z7 + a-3z9 + 3a-2 - 14a-2z2 + 14a-2z4 - 6a-2z6 + 2a-2z8 + 2a-1z - 5a-1z3 + a-1z5 + a-1z7 + 2 - 6z2 + 3z4

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

{1, 4}

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=2 is the signature of 1129. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 r = 7 j = 17                   1 j = 15                 1   j = 13               3 1   j = 11             4 1     j = 9           4 3       j = 7         5 4         j = 5       3 4           j = 3     3 5             j = 1   2 4               j = -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, 29]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 29]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 13, 6, 12], X[2, 8, 3, 7], > X[14, 9, 15, 10], X[18, 12, 19, 11], X[13, 7, 14, 6], X[22, 15, 1, 16], > X[20, 17, 21, 18], X[10, 20, 11, 19], X[16, 21, 17, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 29]]
Out[4]=	GaussCode[1, -4, 2, -1, -3, 7, 4, -2, 5, -10, 6, 3, -7, -5, 8, -11, 9, -6, 10, > -9, 11, -8]
In[5]:=	DTCode[Knot[11, NonAlternating, 29]]
Out[5]=	DTCode[4, 8, -12, 2, 14, 18, -6, 22, 20, 10, 16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 29]][t]
Out[6]=	3 13 2 -19 - -- + -- + 13 t - 3 t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 29]][z]
Out[7]=	2 4 1 + z - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 36], Knot[11, Alternating, 230], Knot[11, NonAlternating, 29]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 29]], KnotSignature[Knot[11, NonAlternating, 29]]}
Out[9]=	{51, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 29]][q]
Out[10]=	2 2 3 4 5 6 7 8 -4 + - + 6 q - 8 q + 9 q - 8 q + 7 q - 4 q + 2 q - q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 29]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 29]][q]
Out[12]=	2 2 4 8 12 16 20 22 24 26 -- + 2 q - 3 q - q + 2 q + 3 q - q + q - q - q 4 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 29]][a, z]
Out[13]=	2 2 2 4 4 -8 -6 2 3 2 2 z z 4 z z 2 z 2 - a + a + -- - -- + 2 z + ---- + -- - ---- - -- - ---- 4 2 6 4 2 4 2 a a a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 29]][a, z]
Out[14]=	2 2 -8 -6 2 3 2 z 2 z z 3 z 2 z 2 3 z 11 z 2 - a - a + -- + -- + --- + --- + -- + --- + --- - 6 z + ---- - ----- - 4 2 9 7 5 3 a 8 4 a a a a a a a a 2 3 3 3 3 4 4 4 4 14 z 3 z 3 z 5 z 5 z 4 5 z 2 z 18 z 14 z > ----- - ---- - ---- - ---- - ---- + 3 z - ---- + ---- + ----- + ----- + 2 9 7 3 a 8 6 4 2 a a a a a a a a 5 5 5 5 6 6 6 6 7 7 7 z 2 z 4 z z 2 z 3 z 11 z 6 z 2 z z z > -- - ---- + ---- + -- + ---- - ---- - ----- - ---- + ---- - -- + -- + 9 7 3 a 8 6 4 2 7 3 a a a a a a a a a a 8 8 8 9 9 2 z 4 z 2 z z z > ---- + ---- + ---- + -- + -- 6 4 2 5 3 a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 29]], Vassiliev[3][Knot[11, NonAlternating, 29]]}
Out[15]=	{1, 4}
In[16]:=	Kh[Knot[11, NonAlternating, 29]][q, t]
Out[16]=	3 2 2 2 q 3 5 5 2 7 2 4 q + 3 q + ----- + --- + --- + 5 q t + 3 q t + 4 q t + 5 q t + 3 2 q t t q t 7 3 9 3 9 4 11 4 11 5 13 5 13 6 > 4 q t + 4 q t + 3 q t + 4 q t + q t + 3 q t + q t + 15 6 17 7 > q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n29

K11n28 K11n30