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

K11n121 K11n123

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X5,16,6,17 X7,12,8,13 X9,19,10,18 X2,11,3,12 X13,20,14,21 X15,6,16,7 X17,22,18,1 X19,9,20,8 X21,14,22,15

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- 2t-2 + 7t-1 - 9 + 7t - 2t2

Conway Polynomial:

1 - z2 - 2z4

Other knots with the same Alexander/Conway Polynomial:

{811, 10147, ...}

Determinant and Signature:

{27, -2}

Jones Polynomial:

q-9 - 2q-8 + 3q-7 - 4q-6 + 4q-5 - 5q-4 + 4q-3 - 2q-2 + 2q-1

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

{...}

A2 (sl(3)) Invariant:

q-28 + q-22 - q-20 - q-16 - 2q-14 - q-12 - q-10 + 2q-8 + 2q-6 + q-4 + 2q-2

HOMFLY-PT Polynomial:

3a2 + 2a2z2 - 2a4 - 2a4z2 - a4z4 - a6 - 2a6z2 - a6z4 + a8 + a8z2

Kauffman Polynomial:

- 3a2 + 2a2z2 + 3a3z - 2a3z3 + a3z5 - 2a4 - 2a4z2 + 9a4z4 - 5a4z6 + a4z8 + 5a5z - 9a5z3 + 10a5z5 - 5a5z7 + a5z9 + a6 - 11a6z2 + 21a6z4 - 14a6z6 + 3a6z8 + a7z5 - 3a7z7 + a7z9 + a8 - 4a8z2 + 8a8z4 - 8a8z6 + 2a8z8 - 2a9z + 7a9z3 - 8a9z5 + 2a9z7 + 3a10z2 - 4a10z4 + a10z6

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

{-1, 4}

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 11122. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -1                 2 j = -3               1 1 j = -5             3 1   j = -7           2 1     j = -9         2 3       j = -11       2 2         j = -13     1 2           j = -15   1 2             j = -17   1               j = -19 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, 122]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 122]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[5, 16, 6, 17], X[7, 12, 8, 13], > X[9, 19, 10, 18], X[2, 11, 3, 12], X[13, 20, 14, 21], X[15, 6, 16, 7], > X[17, 22, 18, 1], X[19, 9, 20, 8], X[21, 14, 22, 15]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 122]]
Out[4]=	GaussCode[1, -6, 2, -1, -3, 8, -4, 10, -5, -2, 6, 4, -7, 11, -8, 3, -9, 5, -10, > 7, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 122]]
Out[5]=	DTCode[4, 10, -16, -12, -18, 2, -20, -6, -22, -8, -14]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 122]][t]
Out[6]=	2 7 2 -9 - -- + - + 7 t - 2 t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 122]][z]
Out[7]=	2 4 1 - z - 2 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[8, 11], Knot[10, 147], Knot[11, NonAlternating, 122]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 122]], KnotSignature[Knot[11, NonAlternating, 122]]}
Out[9]=	{27, -2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 122]][q]
Out[10]=	-9 2 3 4 4 5 4 2 2 q - -- + -- - -- + -- - -- + -- - -- + - 8 7 6 5 4 3 2 q q q q q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[11, NonAlternating, 122]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 122]][q]
Out[12]=	-28 -22 -20 -16 2 -12 -10 2 2 -4 2 q + q - q - q - --- - q - q + -- + -- + q + -- 14 8 6 2 q q q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 122]][a, z]
Out[13]=	2 4 6 8 2 2 4 2 6 2 8 2 4 4 6 4 3 a - 2 a - a + a + 2 a z - 2 a z - 2 a z + a z - a z - a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 122]][a, z]
Out[14]=	2 4 6 8 3 5 9 2 2 4 2 -3 a - 2 a + a + a + 3 a z + 5 a z - 2 a z + 2 a z - 2 a z - 6 2 8 2 10 2 3 3 5 3 9 3 4 4 > 11 a z - 4 a z + 3 a z - 2 a z - 9 a z + 7 a z + 9 a z + 6 4 8 4 10 4 3 5 5 5 7 5 9 5 > 21 a z + 8 a z - 4 a z + a z + 10 a z + a z - 8 a z - 4 6 6 6 8 6 10 6 5 7 7 7 9 7 > 5 a z - 14 a z - 8 a z + a z - 5 a z - 3 a z + 2 a z + 4 8 6 8 8 8 5 9 7 9 > a z + 3 a z + 2 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 122]], Vassiliev[3][Knot[11, NonAlternating, 122]]}
Out[15]=	{-1, 4}
In[16]:=	Kh[Knot[11, NonAlternating, 122]][q, t]
Out[16]=	-3 2 1 1 1 2 1 2 2 q + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + q 19 8 17 7 15 7 15 6 13 6 13 5 11 5 q t q t q t q t q t q t q t 2 2 3 2 1 3 1 1 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- 11 4 9 4 9 3 7 3 7 2 5 2 5 3 q t q t q t q t q t q t q t q t

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n122

K11n121 K11n123