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

K11a122 K11a124

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 X8,12,9,11 X18,14,19,13 X22,16,1,15 X6,18,7,17 X12,20,13,19 X16,22,17,21

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

4 10 14 20 2 8 18 22 6 12 16

Alexander Polynomial:

9t-2 - 29t-1 + 41 - 29t + 9t2

Conway Polynomial:

1 + 7z2 + 9z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{117, 4}

Jones Polynomial:

q2 - 3q3 + 7q4 - 11q5 + 17q6 - 18q7 + 18q8 - 17q9 + 12q10 - 8q11 + 4q12 - q13

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

{...}

A2 (sl(3)) Invariant:

q6 - 2q8 + 2q10 + q12 - 3q14 + 5q16 + q18 + q20 + 3q22 - 2q24 + 2q26 - 4q28 - 2q30 + q32 - 4q34 + 2q36 + 2q38 - q40

HOMFLY-PT Polynomial:

a-12 - a-12z2 - 4a-10 - 3a-10z2 + a-10z4 + 2a-8 + 6a-8z2 + 4a-8z4 + 2a-6 + 4a-6z2 + 3a-6z4 + a-4z2 + a-4z4

Kauffman Polynomial:

- a-15z + 3a-15z3 - 3a-15z5 + a-15z7 - 3a-14z2 + 14a-14z4 - 14a-14z6 + 4a-14z8 - 4a-13z + 6a-13z3 + 6a-13z5 - 14a-13z7 + 5a-13z9 + a-12 - 6a-12z2 + 31a-12z4 - 34a-12z6 + 6a-12z8 + 2a-12z10 - 12a-11z + 20a-11z3 + a-11z5 - 25a-11z7 + 11a-11z9 + 4a-10 - 12a-10z2 + 33a-10z4 - 40a-10z6 + 11a-10z8 + 2a-10z10 - 9a-9z + 23a-9z3 - 20a-9z5 - 2a-9z7 + 6a-9z9 + 2a-8 - 2a-8z2 + 8a-8z4 - 14a-8z6 + 9a-8z8 + 4a-7z3 - 9a-7z5 + 8a-7z7 - 2a-6 + 6a-6z2 - 7a-6z4 + 6a-6z6 - 2a-5z3 + 3a-5z5 - a-4z2 + a-4z4

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

{7, 17}

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 11123. 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 = 27                       1 j = 25                     3   j = 23                   5 1   j = 21                 7 3     j = 19               10 5       j = 17             8 7         j = 15           10 10           j = 13         7 8             j = 11       4 10               j = 9     3 7                 j = 7     4                   j = 5 1 3                     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, Alternating, 123]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, Alternating, 123]]
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[8, 12, 9, 11], X[18, 14, 19, 13], X[22, 16, 1, 15], > X[6, 18, 7, 17], X[12, 20, 13, 19], X[16, 22, 17, 21]]
In[4]:=	GaussCode[Knot[11, Alternating, 123]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -9, 4, -6, 5, -2, 6, -10, 7, -3, 8, -11, 9, -7, 10, > -4, 11, -8]
In[5]:=	DTCode[Knot[11, Alternating, 123]]
Out[5]=	DTCode[4, 10, 14, 20, 2, 8, 18, 22, 6, 12, 16]
In[6]:=	alex = Alexander[Knot[11, Alternating, 123]][t]
Out[6]=	9 29 2 41 + -- - -- - 29 t + 9 t 2 t t
In[7]:=	Conway[Knot[11, Alternating, 123]][z]
Out[7]=	2 4 1 + 7 z + 9 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, Alternating, 123]}
In[9]:=	{KnotDet[Knot[11, Alternating, 123]], KnotSignature[Knot[11, Alternating, 123]]}
Out[9]=	{117, 4}
In[10]:=	J=Jones[Knot[11, Alternating, 123]][q]
Out[10]=	2 3 4 5 6 7 8 9 10 11 q - 3 q + 7 q - 11 q + 17 q - 18 q + 18 q - 17 q + 12 q - 8 q + 12 13 > 4 q - q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, Alternating, 123]}
In[12]:=	A2Invariant[Knot[11, Alternating, 123]][q]
Out[12]=	6 8 10 12 14 16 18 20 22 24 26 q - 2 q + 2 q + q - 3 q + 5 q + q + q + 3 q - 2 q + 2 q - 28 30 32 34 36 38 40 > 4 q - 2 q + q - 4 q + 2 q + 2 q - q
In[13]:=	HOMFLYPT[Knot[11, Alternating, 123]][a, z]
Out[13]=	2 2 2 2 2 4 4 4 4 -12 4 2 2 z 3 z 6 z 4 z z z 4 z 3 z z a - --- + -- + -- - --- - ---- + ---- + ---- + -- + --- + ---- + ---- + -- 10 8 6 12 10 8 6 4 10 8 6 4 a a a a a a a a a a a a
In[14]:=	Kauffman[Knot[11, Alternating, 123]][a, z]
Out[14]=	2 2 2 2 -12 4 2 2 z 4 z 12 z 9 z 3 z 6 z 12 z 2 z a + --- + -- - -- - --- - --- - ---- - --- - ---- - ---- - ----- - ---- + 10 8 6 15 13 11 9 14 12 10 8 a a a a a a a a a a a 2 2 3 3 3 3 3 3 4 4 6 z z 3 z 6 z 20 z 23 z 4 z 2 z 14 z 31 z > ---- - -- + ---- + ---- + ----- + ----- + ---- - ---- + ----- + ----- + 6 4 15 13 11 9 7 5 14 12 a a a a a a a a a a 4 4 4 4 5 5 5 5 5 5 33 z 8 z 7 z z 3 z 6 z z 20 z 9 z 3 z > ----- + ---- - ---- + -- - ---- + ---- + --- - ----- - ---- + ---- - 10 8 6 4 15 13 11 9 7 5 a a a a a a a a a a 6 6 6 6 6 7 7 7 7 7 14 z 34 z 40 z 14 z 6 z z 14 z 25 z 2 z 8 z > ----- - ----- - ----- - ----- + ---- + --- - ----- - ----- - ---- + ---- + 14 12 10 8 6 15 13 11 9 7 a a a a a a a a a a 8 8 8 8 9 9 9 10 10 4 z 6 z 11 z 9 z 5 z 11 z 6 z 2 z 2 z > ---- + ---- + ----- + ---- + ---- + ----- + ---- + ----- + ----- 14 12 10 8 13 11 9 12 10 a a a a a a a a a
In[15]:=	{Vassiliev[2][Knot[11, Alternating, 123]], Vassiliev[3][Knot[11, Alternating, 123]]}
Out[15]=	{7, 17}
In[16]:=	Kh[Knot[11, Alternating, 123]][q, t]
Out[16]=	3 5 5 7 2 9 2 9 3 11 3 11 4 q + q + 3 q t + 4 q t + 3 q t + 7 q t + 4 q t + 10 q t + 13 4 13 5 15 5 15 6 17 6 17 7 > 7 q t + 8 q t + 10 q t + 10 q t + 8 q t + 7 q t + 19 7 19 8 21 8 21 9 23 9 23 10 > 10 q t + 5 q t + 7 q t + 3 q t + 5 q t + q t + 25 10 27 11 > 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11a123

K11a122 K11a124