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

K11n123 K11n125

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X16,6,17,5 X7,15,8,14 X9,19,10,18 X2,11,3,12 X13,9,14,8 X20,15,21,16 X22,18,1,17 X19,13,20,12 X6,21,7,22

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

t-3 - 6t-2 + 14t-1 - 17 + 14t - 6t2 + t3

Conway Polynomial:

1 - z2 + z6

Other knots with the same Alexander/Conway Polynomial:

{932, K11n52, ...}

Determinant and Signature:

{59, 2}

Jones Polynomial:

q-3 - 3q-2 + 6q-1 - 8 + 10q - 10q2 + 9q3 - 7q4 + 4q5 - q6

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

{K11n166, ...}

A2 (sl(3)) Invariant:

q-10 - q-6 + 2q-4 - q-2 + 1 + q2 - 2q4 + 2q6 - 2q8 + 2q10 - q14 + 2q16 - q18

HOMFLY-PT Polynomial:

- a-4z2 - a-4z4 + a-2 + 3a-2z2 + 3a-2z4 + a-2z6 - 1 - 4z2 - 2z4 + a2 + a2z2

Kauffman Polynomial:

a-7z3 - a-6z2 + 4a-6z4 - a-5z3 + 3a-5z5 + a-5z7 - a-4z2 + 2a-4z4 - a-4z6 + 2a-4z8 + a-3z3 - 6a-3z5 + 3a-3z7 + a-3z9 - a-2 + 4a-2z2 - 4a-2z4 - 7a-2z6 + 5a-2z8 - a-1z + 10a-1z3 - 18a-1z5 + 5a-1z7 + a-1z9 - 1 + 7z2 - 5z4 - 5z6 + 3z8 - az + 7az3 - 9az5 + 3az7 - a2 + 3a2z2 - 3a2z4 + a2z6

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

{-1, 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=2 is the signature of 11124. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 r = 4 r = 5 j = 13                   1 j = 11                 3   j = 9               4 1   j = 7             5 3     j = 5           5 4       j = 3         5 5         j = 1       4 6           j = -1     2 4             j = -3   1 4               j = -5   2                 j = -7 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, 124]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 124]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[16, 6, 17, 5], X[7, 15, 8, 14], > X[9, 19, 10, 18], X[2, 11, 3, 12], X[13, 9, 14, 8], X[20, 15, 21, 16], > X[22, 18, 1, 17], X[19, 13, 20, 12], X[6, 21, 7, 22]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 124]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -11, -4, 7, -5, -2, 6, 10, -7, 4, 8, -3, 9, 5, -10, > -8, 11, -9]
In[5]:=	DTCode[Knot[11, NonAlternating, 124]]
Out[5]=	DTCode[4, 10, 16, -14, -18, 2, -8, 20, 22, -12, 6]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 124]][t]
Out[6]=	-3 6 14 2 3 -17 + t - -- + -- + 14 t - 6 t + t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 124]][z]
Out[7]=	2 6 1 - z + z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 32], Knot[11, NonAlternating, 52], Knot[11, NonAlternating, 124]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 124]], KnotSignature[Knot[11, NonAlternating, 124]]}
Out[9]=	{59, 2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 124]][q]
Out[10]=	-3 3 6 2 3 4 5 6 -8 + q - -- + - + 10 q - 10 q + 9 q - 7 q + 4 q - q 2 q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 124], Knot[11, NonAlternating, 166]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 124]][q]
Out[12]=	-10 -6 2 -2 2 4 6 8 10 14 16 18 1 + q - q + -- - q + q - 2 q + 2 q - 2 q + 2 q - q + 2 q - q 4 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 124]][a, z]
Out[13]=	2 2 4 4 6 -2 2 2 z 3 z 2 2 4 z 3 z z -1 + a + a - 4 z - -- + ---- + a z - 2 z - -- + ---- + -- 4 2 4 2 2 a a a a a
In[14]:=	Kauffman[Knot[11, NonAlternating, 124]][a, z]
Out[14]=	2 2 2 3 3 3 -2 2 z 2 z z 4 z 2 2 z z z -1 - a - a - - - a z + 7 z - -- - -- + ---- + 3 a z + -- - -- + -- + a 6 4 2 7 5 3 a a a a a a 3 4 4 4 5 5 10 z 3 4 4 z 2 z 4 z 2 4 3 z 6 z > ----- + 7 a z - 5 z + ---- + ---- - ---- - 3 a z + ---- - ---- - a 6 4 2 5 3 a a a a a 5 6 6 7 7 7 18 z 5 6 z 7 z 2 6 z 3 z 5 z 7 > ----- - 9 a z - 5 z - -- - ---- + a z + -- + ---- + ---- + 3 a z + a 4 2 5 3 a a a a a 8 8 9 9 8 2 z 5 z z z > 3 z + ---- + ---- + -- + -- 4 2 3 a a a a
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 124]], Vassiliev[3][Knot[11, NonAlternating, 124]]}
Out[15]=	{-1, 0}
In[16]:=	Kh[Knot[11, NonAlternating, 124]][q, t]
Out[16]=	3 1 2 1 4 2 4 4 q 3 6 q + 5 q + ----- + ----- + ----- + ----- + ---- + --- + --- + 5 q t + 7 4 5 3 3 3 3 2 2 q t t q t q t q t q t q t 5 5 2 7 2 7 3 9 3 9 4 11 4 13 5 > 5 q t + 4 q t + 5 q t + 3 q t + 4 q t + q t + 3 q t + q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n124

K11n123 K11n125