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

K11a23 K11a25

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8493 X12,5,13,6 X2837 X18,10,19,9 X16,11,17,12 X6,13,7,14 X20,15,21,16 X22,18,1,17 X14,19,15,20 X10,22,11,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 8 12 2 18 16 6 20 22 14 10

Alexander Polynomial:

t-4 - 6t-3 + 18t-2 - 33t-1 + 41 - 33t + 18t2 - 6t3 + t4

Conway Polynomial:

1 + z2 + 2z4 + 2z6 + z8

Other knots with the same Alexander/Conway Polynomial:

{K11a26, K11a315, ...}

Determinant and Signature:

{157, 0}

Jones Polynomial:

- q-5 + 4q-4 - 10q-3 + 17q-2 - 22q-1 + 26 - 25q + 22q2 - 16q3 + 9q4 - 4q5 + q6

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

{K11a26, K11a315, ...}

A2 (sl(3)) Invariant:

- q-14 + 2q-12 - 4q-10 + 2q-8 + q-6 - 3q-4 + 7q-2 - 3 + 5q2 - q4 - 2q6 + 3q8 - 5q10 + 2q12 - q16 + q18

HOMFLY-PT Polynomial:

a-4 + 2a-4z2 + a-4z4 - 4a-2 - 9a-2z2 - 7a-2z4 - 2a-2z6 + 6 + 12z2 + 11z4 + 5z6 + z8 - 2a2 - 4a2z2 - 3a2z4 - a2z6

Kauffman Polynomial:

a-6z2 - 2a-6z4 + a-6z6 - 2a-5z + 7a-5z3 - 9a-5z5 + 4a-5z7 + a-4 - 3a-4z2 + 9a-4z4 - 14a-4z6 + 7a-4z8 - 7a-3z + 22a-3z3 - 21a-3z5 - a-3z7 + 6a-3z9 + 4a-2 - 18a-2z2 + 42a-2z4 - 50a-2z6 + 18a-2z8 + 2a-2z10 - 10a-1z + 30a-1z3 - 24a-1z5 - 12a-1z7 + 14a-1z9 + 6 - 23z2 + 50z4 - 58z6 + 23z8 + 2z10 - 8az + 25az3 - 26az5 + 2az7 + 8az9 + 2a2 - 8a2z2 + 15a2z4 - 19a2z6 + 12a2z8 - 3a3z + 9a3z3 - 13a3z5 + 9a3z7 + a4z2 - 4a4z4 + 4a4z6 - a5z3 + a5z5

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

{1, -1}

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=0 is the signature of 1124. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 r = 6 j = 13                       1 j = 11                     3   j = 9                   6 1   j = 7                 10 3     j = 5               12 6       j = 3             13 10         j = 1           13 12           j = -1         10 14             j = -3       7 12               j = -5     3 10                 j = -7   1 7                   j = -9   3                     j = -11 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, Alternating, 24]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 24]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 4, 9, 3], X[12, 5, 13, 6], X[2, 8, 3, 7], > X[18, 10, 19, 9], X[16, 11, 17, 12], X[6, 13, 7, 14], X[20, 15, 21, 16], > X[22, 18, 1, 17], X[14, 19, 15, 20], X[10, 22, 11, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 24]]
Out[4]=	GaussCode[1, -4, 2, -1, 3, -7, 4, -2, 5, -11, 6, -3, 7, -10, 8, -6, 9, -5, 10, > -8, 11, -9]
In[5]:=	DTCode[Knot[11, Alternating, 24]]
Out[5]=	DTCode[4, 8, 12, 2, 18, 16, 6, 20, 22, 14, 10]
In[6]:=	alex = Alexander[Knot[11, Alternating, 24]][t]
Out[6]=	-4 6 18 33 2 3 4 41 + t - -- + -- - -- - 33 t + 18 t - 6 t + t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 24]][z]
Out[7]=	2 4 6 8 1 + z + 2 z + 2 z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 24], Knot[11, Alternating, 26], > Knot[11, Alternating, 315]}
In[9]:=	{KnotDet[Knot[11, Alternating, 24]], KnotSignature[Knot[11, Alternating, 24]]}
Out[9]=	{157, 0}
In[10]:=	J=Jones[Knot[11, Alternating, 24]][q]
Out[10]=	-5 4 10 17 22 2 3 4 5 6 26 - q + -- - -- + -- - -- - 25 q + 22 q - 16 q + 9 q - 4 q + q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 24], Knot[11, Alternating, 26], > Knot[11, Alternating, 315]}
In[12]:=	A2Invariant[Knot[11, Alternating, 24]][q]
Out[12]=	-14 2 4 2 -6 3 7 2 4 6 8 10 -3 - q + --- - --- + -- + q - -- + -- + 5 q - q - 2 q + 3 q - 5 q + 12 10 8 4 2 q q q q q 12 16 18 > 2 q - q + q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 24]][a, z]
Out[13]=	2 2 4 4 -4 4 2 2 2 z 9 z 2 2 4 z 7 z 6 + a - -- - 2 a + 12 z + ---- - ---- - 4 a z + 11 z + -- - ---- - 2 4 2 4 2 a a a a a 6 2 4 6 2 z 2 6 8 > 3 a z + 5 z - ---- - a z + z 2 a
In[14]:=	Kauffman[Knot[11, Alternating, 24]][a, z]
Out[14]=	2 2 -4 4 2 2 z 7 z 10 z 3 2 z 3 z 6 + a + -- + 2 a - --- - --- - ---- - 8 a z - 3 a z - 23 z + -- - ---- - 2 5 3 a 6 4 a a a a a 2 3 3 3 18 z 2 2 4 2 7 z 22 z 30 z 3 3 3 > ----- - 8 a z + a z + ---- + ----- + ----- + 25 a z + 9 a z - 2 5 3 a a a a 4 4 4 5 5 5 3 4 2 z 9 z 42 z 2 4 4 4 9 z 21 z > a z + 50 z - ---- + ---- + ----- + 15 a z - 4 a z - ---- - ----- - 6 4 2 5 3 a a a a a 5 6 6 6 24 z 5 3 5 5 5 6 z 14 z 50 z > ----- - 26 a z - 13 a z + a z - 58 z + -- - ----- - ----- - a 6 4 2 a a a 7 7 7 8 2 6 4 6 4 z z 12 z 7 3 7 8 7 z > 19 a z + 4 a z + ---- - -- - ----- + 2 a z + 9 a z + 23 z + ---- + 5 3 a 4 a a a 8 9 9 10 18 z 2 8 6 z 14 z 9 10 2 z > ----- + 12 a z + ---- + ----- + 8 a z + 2 z + ----- 2 3 a 2 a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 24]], Vassiliev[3][Knot[11, Alternating, 24]]}
Out[15]=	{1, -1}
In[16]:=	Kh[Knot[11, Alternating, 24]][q, t]
Out[16]=	14 1 3 1 7 3 10 7 12 -- + 13 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + ---- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 3 q t q t q t q t q t q t q t q t 10 3 3 2 5 2 5 3 7 3 > --- + 12 q t + 13 q t + 10 q t + 12 q t + 6 q t + 10 q t + q t 7 4 9 4 9 5 11 5 13 6 > 3 q t + 6 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a24

K11a23 K11a25