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

K11n29 K11n31

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X5,13,6,12 X2837 X9,15,10,14 X11,18,12,19 X13,7,14,6 X15,1,16,22 X17,21,18,20 X19,10,20,11 X21,17,22,16

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:

- 2t-3 + 6t-2 - 6t-1 + 5 - 6t + 6t2 - 2t3

Conway Polynomial:

1 - 6z4 - 2z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{33, 4}

Jones Polynomial:

1 - q + 3q2 - 4q3 + 5q4 - 6q5 + 5q6 - 4q7 + 3q8 - q9

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

{...}

A2 (sl(3)) Invariant:

1 + q2 + q4 + 2q6 + q10 - 2q12 - q14 - q16 - 2q18 + q20 + 2q24 + q26 - q32

HOMFLY-PT Polynomial:

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

Kauffman Polynomial:

- a-11z + a-11z3 + a-10 - 4a-10z2 + 3a-10z4 - a-9z + 4a-9z3 - 2a-9z5 + a-9z7 + 3a-8 - 17a-8z2 + 23a-8z4 - 10a-8z6 + 2a-8z8 + 3a-7z - 8a-7z3 + 9a-7z5 - 4a-7z7 + a-7z9 + 2a-6 - 15a-6z2 + 22a-6z4 - 13a-6z6 + 3a-6z8 + 5a-5z - 11a-5z3 + 8a-5z5 - 4a-5z7 + a-5z9 - 2a-4 + 5a-4z2 - 3a-4z4 - 2a-4z6 + a-4z8 + 2a-3z - 3a-3z5 + a-3z7 - 3a-2 + 7a-2z2 - 5a-2z4 + a-2z6

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

{0, 0}

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 1130. 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 = 19                   1 j = 17                 2   j = 15               2 1   j = 13             3 2     j = 11           3 2       j = 9         2 3         j = 7       2 3           j = 5     1 2             j = 3   1 3               j = 1                     j = -1 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]:=	Show[DrawMorseLink[Knot[11, NonAlternating, 30]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 30]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[5, 13, 6, 12], X[2, 8, 3, 7], > X[9, 15, 10, 14], X[11, 18, 12, 19], X[13, 7, 14, 6], X[15, 1, 16, 22], > X[17, 21, 18, 20], X[19, 10, 20, 11], X[21, 17, 22, 16]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 30]]
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, 30]]
Out[5]=	DTCode[4, 8, -12, 2, -14, -18, -6, -22, -20, -10, -16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 30]][t]
Out[6]=	2 6 6 2 3 5 - -- + -- - - - 6 t + 6 t - 2 t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 30]][z]
Out[7]=	4 6 1 - 6 z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 30]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 30]], KnotSignature[Knot[11, NonAlternating, 30]]}
Out[9]=	{33, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 30]][q]
Out[10]=	2 3 4 5 6 7 8 9 1 - q + 3 q - 4 q + 5 q - 6 q + 5 q - 4 q + 3 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 30]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 30]][q]
Out[12]=	2 4 6 10 12 14 16 18 20 24 26 32 1 + q + q + 2 q + q - 2 q - q - q - 2 q + q + 2 q + q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 30]][a, z]
Out[13]=	2 2 2 2 4 4 4 4 -10 3 2 2 3 4 z 4 z 4 z 4 z z 4 z 4 z z -a + -- - -- - -- + -- + ---- - ---- - ---- + ---- + -- - ---- - ---- + -- - 8 6 4 2 8 6 4 2 8 6 4 2 a a a a a a a a a a a a 6 6 z z > -- - -- 6 4 a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 30]][a, z]
Out[14]=	2 2 2 -10 3 2 2 3 z z 3 z 5 z 2 z 4 z 17 z 15 z a + -- + -- - -- - -- - --- - -- + --- + --- + --- - ---- - ----- - ----- + 8 6 4 2 11 9 7 5 3 10 8 6 a a a a a a a a a a a a 2 2 3 3 3 3 4 4 4 4 5 z 7 z z 4 z 8 z 11 z 3 z 23 z 22 z 3 z > ---- + ---- + --- + ---- - ---- - ----- + ---- + ----- + ----- - ---- - 4 2 11 9 7 5 10 8 6 4 a a a a a a a a a a 4 5 5 5 5 6 6 6 6 7 7 5 z 2 z 9 z 8 z 3 z 10 z 13 z 2 z z z 4 z > ---- - ---- + ---- + ---- - ---- - ----- - ----- - ---- + -- + -- - ---- - 2 9 7 5 3 8 6 4 2 9 7 a a a a a a a a a a a 7 7 8 8 8 9 9 4 z z 2 z 3 z z z z > ---- + -- + ---- + ---- + -- + -- + -- 5 3 8 6 4 7 5 a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 30]], Vassiliev[3][Knot[11, NonAlternating, 30]]}
Out[15]=	{0, 0}
In[16]:=	Kh[Knot[11, NonAlternating, 30]][q, t]
Out[16]=	3 3 5 1 q 5 7 7 2 9 2 9 3 3 q + q + ---- + -- + 2 q t + 2 q t + 3 q t + 2 q t + 3 q t + 2 t q t 11 3 11 4 13 4 13 5 15 5 15 6 17 6 > 3 q t + 2 q t + 3 q t + 2 q t + 2 q t + q t + 2 q t + 19 7 > q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n30

K11n29 K11n31