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

K11n122 K11n124

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X5,16,6,17 X7,15,8,14 X12,10,13,9 X2,11,3,12 X13,18,14,19 X15,21,16,20 X17,22,18,1 X19,8,20,9 X21,7,22,6

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

3t-2 - 14t-1 + 23 - 14t + 3t2

Conway Polynomial:

1 - 2z2 + 3z4

Other knots with the same Alexander/Conway Polynomial:

{...}

Determinant and Signature:

{57, 0}

Jones Polynomial:

q-6 - 3q-5 + 6q-4 - 8q-3 + 9q-2 - 10q-1 + 9 - 6q + 4q2 - q3

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

{...}

A2 (sl(3)) Invariant:

q-20 + q-18 - 2q-16 + q-14 - 2q-10 + 2q-8 - q-6 + q-4 - 1 + 2q2 - 2q4 + 2q6 + 2q8 - q10

HOMFLY-PT Polynomial:

a-2 - a-2z2 - 1 - z2 + z4 + 2a2 + 3a2z2 + 2a2z4 - 2a4 - 3a4z2 + a6

Kauffman Polynomial:

a-3z3 - a-2 - 2a-2z2 + 4a-2z4 + 2a-1z5 + a-1z7 - 1 + 3z4 - 2z6 + 2z8 + 6az3 - 9az5 + 3az7 + az9 - 2a2 + 7a2z2 - 4a2z4 - 8a2z6 + 5a2z8 - 2a3z + 14a3z3 - 20a3z5 + 5a3z7 + a3z9 - 2a4 + 8a4z2 - 6a4z4 - 5a4z6 + 3a4z8 - 2a5z + 7a5z3 - 9a5z5 + 3a5z7 - a6 + 3a6z2 - 3a6z4 + a6z6

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

{-2, 2}

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 11123. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 r = 3 j = 7                   1 j = 5                 3   j = 3               3 1   j = 1             6 3     j = -1           5 4       j = -3         4 5         j = -5       4 5           j = -7     2 4             j = -9   1 4               j = -11   2                 j = -13 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, 123]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 123]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[5, 16, 6, 17], X[7, 15, 8, 14], > X[12, 10, 13, 9], X[2, 11, 3, 12], X[13, 18, 14, 19], X[15, 21, 16, 20], > X[17, 22, 18, 1], X[19, 8, 20, 9], X[21, 7, 22, 6]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 123]]
Out[4]=	GaussCode[1, -6, 2, -1, -3, 11, -4, 10, 5, -2, 6, -5, -7, 4, -8, 3, -9, 7, -10, > 8, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 123]]
Out[5]=	DTCode[4, 10, -16, -14, 12, 2, -18, -20, -22, -8, -6]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 123]][t]
Out[6]=	3 14 2 23 + -- - -- - 14 t + 3 t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 123]][z]
Out[7]=	2 4 1 - 2 z + 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 123]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 123]], KnotSignature[Knot[11, NonAlternating, 123]]}
Out[9]=	{57, 0}
In[10]:=	J=Jones[Knot[11, NonAlternating, 123]][q]
Out[10]=	-6 3 6 8 9 10 2 3 9 + q - -- + -- - -- + -- - -- - 6 q + 4 q - q 5 4 3 2 q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 123]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 123]][q]
Out[12]=	-20 -18 2 -14 2 2 -6 -4 2 4 6 -1 + q + q - --- + q - --- + -- - q + q + 2 q - 2 q + 2 q + 16 10 8 q q q 8 10 > 2 q - q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 123]][a, z]
Out[13]=	2 -2 2 4 6 2 z 2 2 4 2 4 2 4 -1 + a + 2 a - 2 a + a - z - -- + 3 a z - 3 a z + z + 2 a z 2 a
In[14]:=	Kauffman[Knot[11, NonAlternating, 123]][a, z]
Out[14]=	2 -2 2 4 6 3 5 2 z 2 2 4 2 -1 - a - 2 a - 2 a - a - 2 a z - 2 a z - ---- + 7 a z + 8 a z + 2 a 3 4 6 2 z 3 3 3 5 3 4 4 z 2 4 > 3 a z + -- + 6 a z + 14 a z + 7 a z + 3 z + ---- - 4 a z - 3 2 a a 5 4 4 6 4 2 z 5 3 5 5 5 6 2 6 > 6 a z - 3 a z + ---- - 9 a z - 20 a z - 9 a z - 2 z - 8 a z - a 7 4 6 6 6 z 7 3 7 5 7 8 2 8 > 5 a z + a z + -- + 3 a z + 5 a z + 3 a z + 2 z + 5 a z + a 4 8 9 3 9 > 3 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 123]], Vassiliev[3][Knot[11, NonAlternating, 123]]}
Out[15]=	{-2, 2}
In[16]:=	Kh[Knot[11, NonAlternating, 123]][q, t]
Out[16]=	4 1 2 1 4 2 4 4 5 - + 6 q + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 13 6 11 5 9 5 9 4 7 4 7 3 5 3 5 2 q t q t q t q t q t q t q t q t 4 5 5 3 3 2 5 2 7 3 > ----- + ---- + --- + 3 q t + 3 q t + q t + 3 q t + q t 3 2 3 q t q t q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n123

K11n122 K11n124