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

K11n22 K11n24

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

t-4 - 3t-3 + 5t-2 - 4t-1 + 3 - 4t + 5t2 - 3t3 + t4

Conway Polynomial:

1 + 5z2 + 7z4 + 5z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{29, 4}

Jones Polynomial:

- q-1 + 2 - 3q + 5q2 - 4q3 + 5q4 - 4q5 + 3q6 - 2q7

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

{...}

A2 (sl(3)) Invariant:

- q-2 - q2 + q6 + 2q8 + 4q10 + q12 + 3q14 - q16 - q18 - 2q20 - 2q22 - q26 + q28

HOMFLY-PT Polynomial:

a-8 + a-8z2 - 7a-6 - 11a-6z2 - 6a-6z4 - a-6z6 + 10a-4 + 22a-4z2 + 18a-4z4 + 7a-4z6 + a-4z8 - 3a-2 - 7a-2z2 - 5a-2z4 - a-2z6

Kauffman Polynomial:

a-9z + a-8 - a-8z2 + a-8z4 - 6a-7z + 13a-7z3 - 9a-7z5 + 2a-7z7 + 7a-6 - 20a-6z2 + 25a-6z4 - 15a-6z6 + 3a-6z8 - 13a-5z + 27a-5z3 - 16a-5z5 + a-5z9 + 10a-4 - 29a-4z2 + 39a-4z4 - 25a-4z6 + 5a-4z8 - 9a-3z + 21a-3z3 - 12a-3z5 - a-3z7 + a-3z9 + 3a-2 - 10a-2z2 + 15a-2z4 - 10a-2z6 + 2a-2z8 - 3a-1z + 7a-1z3 - 5a-1z5 + a-1z7

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

{5, 8}

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 1123. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 15                 2 j = 13               1   j = 11             3 2   j = 9           2 1     j = 7         2 3       j = 5       3 2         j = 3     1 3           j = 1   1 2             j = -1   1               j = -3 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, 23]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 23]]
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, 19, 12, 18], X[13, 7, 14, 6], X[15, 20, 16, 21], > X[17, 11, 18, 10], X[19, 22, 20, 1], X[21, 16, 22, 17]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 23]]
Out[4]=	GaussCode[1, -4, 2, -1, -3, 7, 4, -2, -5, 9, -6, 3, -7, 5, -8, 11, -9, 6, -10, > 8, -11, 10]
In[5]:=	DTCode[Knot[11, NonAlternating, 23]]
Out[5]=	DTCode[4, 8, -12, 2, -14, -18, -6, -20, -10, -22, -16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 23]][t]
Out[6]=	-4 3 5 4 2 3 4 3 + t - -- + -- - - - 4 t + 5 t - 3 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, NonAlternating, 23]][z]
Out[7]=	2 4 6 8 1 + 5 z + 7 z + 5 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 23]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 23]], KnotSignature[Knot[11, NonAlternating, 23]]}
Out[9]=	{29, 4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 23]][q]
Out[10]=	1 2 3 4 5 6 7 2 - - - 3 q + 5 q - 4 q + 5 q - 4 q + 3 q - 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 23]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 23]][q]
Out[12]=	-2 2 6 8 10 12 14 16 18 20 22 26 -q - q + q + 2 q + 4 q + q + 3 q - q - q - 2 q - 2 q - q + 28 > q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 23]][a, z]
Out[13]=	2 2 2 2 4 4 4 6 -8 7 10 3 z 11 z 22 z 7 z 6 z 18 z 5 z z a - -- + -- - -- + -- - ----- + ----- - ---- - ---- + ----- - ---- - -- + 6 4 2 8 6 4 2 6 4 2 6 a a a a a a a a a a a 6 6 8 7 z z z > ---- - -- + -- 4 2 4 a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 23]][a, z]
Out[14]=	2 2 2 2 -8 7 10 3 z 6 z 13 z 9 z 3 z z 20 z 29 z 10 z a + -- + -- + -- + -- - --- - ---- - --- - --- - -- - ----- - ----- - ----- + 6 4 2 9 7 5 3 a 8 6 4 2 a a a a a a a a a a a 3 3 3 3 4 4 4 4 5 5 13 z 27 z 21 z 7 z z 25 z 39 z 15 z 9 z 16 z > ----- + ----- + ----- + ---- + -- + ----- + ----- + ----- - ---- - ----- - 7 5 3 a 8 6 4 2 7 5 a a a a a a a a a 5 5 6 6 6 7 7 7 8 8 12 z 5 z 15 z 25 z 10 z 2 z z z 3 z 5 z > ----- - ---- - ----- - ----- - ----- + ---- - -- + -- + ---- + ---- + 3 a 6 4 2 7 3 a 6 4 a a a a a a a a 8 9 9 2 z z z > ---- + -- + -- 2 5 3 a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 23]], Vassiliev[3][Knot[11, NonAlternating, 23]]}
Out[15]=	{5, 8}
In[16]:=	Kh[Knot[11, NonAlternating, 23]][q, t]
Out[16]=	3 3 5 1 1 q 2 q q 5 7 7 2 3 q + 3 q + ----- + ---- + -- + --- + -- + 2 q t + 2 q t + 3 q t + 3 3 2 2 t t q t q t t 9 2 9 3 11 3 11 4 13 4 15 5 > 2 q t + q t + 3 q t + 2 q t + q t + 2 q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n23

K11n22 K11n24