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

K11a123 K11a125

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,4,11,3 X14,6,15,5 X20,8,21,7 X2,10,3,9 X16,12,17,11 X8,14,9,13 X22,16,1,15 X12,18,13,17 X6,20,7,19 X18,22,19,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 20 2 16 8 22 12 6 18

Alexander Polynomial:

5t-3 - 18t-2 + 34t-1 - 41 + 34t - 18t2 + 5t3

Conway Polynomial:

1 + 7z2 + 12z4 + 5z6

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{155, 6}

Jones Polynomial:

q3 - 3q4 + 9q5 - 14q6 + 21q7 - 25q8 + 25q9 - 23q10 + 17q11 - 11q12 + 5q13 - q14

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

{...}

A2 (sl(3)) Invariant:

q10 - 2q12 + 4q14 - q16 + q18 + 6q20 - 3q22 + 5q24 - 3q26 - 2q28 - 6q32 + 3q34 - 2q36 + 3q40 - q42

HOMFLY-PT Polynomial:

2a-12 - a-12z4 - 7a-10 - 9a-10z2 - a-10z4 + a-10z6 + 5a-8 + 13a-8z2 + 11a-8z4 + 3a-8z6 + a-6 + 3a-6z2 + 3a-6z4 + a-6z6

Kauffman Polynomial:

a-17z5 - 4a-16z4 + 5a-16z6 + 5a-15z3 - 15a-15z5 + 11a-15z7 + 4a-14z4 - 17a-14z6 + 13a-14z8 - 3a-13z + 17a-13z3 - 27a-13z5 + 4a-13z7 + 8a-13z9 + 2a-12 - 3a-12z2 + 23a-12z4 - 45a-12z6 + 21a-12z8 + 2a-12z10 - 13a-11z + 34a-11z3 - 25a-11z5 - 10a-11z7 + 13a-11z9 + 7a-10 - 19a-10z2 + 34a-10z4 - 38a-10z6 + 14a-10z8 + 2a-10z10 - 10a-9z + 25a-9z3 - 20a-9z5 + 5a-9z9 + 5a-8 - 13a-8z2 + 16a-8z4 - 14a-8z6 + 6a-8z8 + 3a-7z3 - 6a-7z5 + 3a-7z7 - a-6 + 3a-6z2 - 3a-6z4 + a-6z6

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

{7, 16}

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=6 is the signature of 11124. 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 r = 10 r = 11 j = 29                       1 j = 27                     4   j = 25                   7 1   j = 23                 10 4     j = 21               13 7       j = 19             12 10         j = 17           13 13           j = 15         8 12             j = 13       6 13               j = 11     3 8                 j = 9     6                   j = 7 1 3                     j = 5 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, 124]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 124]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 4, 11, 3], X[14, 6, 15, 5], X[20, 8, 21, 7], > X[2, 10, 3, 9], X[16, 12, 17, 11], X[8, 14, 9, 13], X[22, 16, 1, 15], > X[12, 18, 13, 17], X[6, 20, 7, 19], X[18, 22, 19, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 124]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -10, 4, -7, 5, -2, 6, -9, 7, -3, 8, -6, 9, -11, 10, > -4, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 124]]
Out[5]=	DTCode[4, 10, 14, 20, 2, 16, 8, 22, 12, 6, 18]
In[6]:=	alex = Alexander[Knot[11, Alternating, 124]][t]
Out[6]=	5 18 34 2 3 -41 + -- - -- + -- + 34 t - 18 t + 5 t 3 2 t t t
In[7]:=	Conway[Knot[11, Alternating, 124]][z]
Out[7]=	2 4 6 1 + 7 z + 12 z + 5 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 124]}
In[9]:=	{KnotDet[Knot[11, Alternating, 124]], KnotSignature[Knot[11, Alternating, 124]]}
Out[9]=	{155, 6}
In[10]:=	J=Jones[Knot[11, Alternating, 124]][q]
Out[10]=	3 4 5 6 7 8 9 10 11 12 q - 3 q + 9 q - 14 q + 21 q - 25 q + 25 q - 23 q + 17 q - 11 q + 13 14 > 5 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 124]}
In[12]:=	A2Invariant[Knot[11, Alternating, 124]][q]
Out[12]=	10 12 14 16 18 20 22 24 26 28 q - 2 q + 4 q - q + q + 6 q - 3 q + 5 q - 3 q - 2 q - 32 34 36 40 42 > 6 q + 3 q - 2 q + 3 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 124]][a, z]
Out[13]=	2 2 2 4 4 4 4 6 2 7 5 -6 9 z 13 z 3 z z z 11 z 3 z z --- - --- + -- + a - ---- + ----- + ---- - --- - --- + ----- + ---- + --- + 12 10 8 10 8 6 12 10 8 6 10 a a a a a a a a a a a 6 6 3 z z > ---- + -- 8 6 a a
In[14]:=	Kauffman[Knot[11, Alternating, 124]][a, z]
Out[14]=	2 2 2 2 3 2 7 5 -6 3 z 13 z 10 z 3 z 19 z 13 z 3 z 5 z --- + --- + -- - a - --- - ---- - ---- - ---- - ----- - ----- + ---- + ---- + 12 10 8 13 11 9 12 10 8 6 15 a a a a a a a a a a a 3 3 3 3 4 4 4 4 4 4 17 z 34 z 25 z 3 z 4 z 4 z 23 z 34 z 16 z 3 z > ----- + ----- + ----- + ---- - ---- + ---- + ----- + ----- + ----- - ---- + 13 11 9 7 16 14 12 10 8 6 a a a a a a a a a a 5 5 5 5 5 5 6 6 6 6 z 15 z 27 z 25 z 20 z 6 z 5 z 17 z 45 z 38 z > --- - ----- - ----- - ----- - ----- - ---- + ---- - ----- - ----- - ----- - 17 15 13 11 9 7 16 14 12 10 a a a a a a a a a a 6 6 7 7 7 7 8 8 8 8 14 z z 11 z 4 z 10 z 3 z 13 z 21 z 14 z 6 z > ----- + -- + ----- + ---- - ----- + ---- + ----- + ----- + ----- + ---- + 8 6 15 13 11 7 14 12 10 8 a a a a a a a a a a 9 9 9 10 10 8 z 13 z 5 z 2 z 2 z > ---- + ----- + ---- + ----- + ----- 13 11 9 12 10 a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 124]], Vassiliev[3][Knot[11, Alternating, 124]]}
Out[15]=	{7, 16}
In[16]:=	Kh[Knot[11, Alternating, 124]][q, t]
Out[16]=	5 7 7 9 2 11 2 11 3 13 3 13 4 q + q + 3 q t + 6 q t + 3 q t + 8 q t + 6 q t + 13 q t + 15 4 15 5 17 5 17 6 19 6 19 7 > 8 q t + 12 q t + 13 q t + 13 q t + 12 q t + 10 q t + 21 7 21 8 23 8 23 9 25 9 25 10 > 13 q t + 7 q t + 10 q t + 4 q t + 7 q t + q t + 27 10 29 11 > 4 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a124

K11a123 K11a125