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

K11a367 K11n2

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X8394 X10,6,11,5 X14,7,15,8 X2,9,3,10 X11,16,12,17 X13,20,14,21 X6,15,7,16 X17,22,18,1 X19,12,20,13 X21,18,22,19

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-2 + 7t-1 - 11 + 7t - t2

Conway Polynomial:

1 + 3z2 - z4

Other knots with the same Alexander/Conway Polynomial:

{948, ...}

Determinant and Signature:

{27, -2}

Jones Polynomial:

- q-10 + 2q-9 - 3q-8 + 4q-7 - 4q-6 + 4q-5 - 4q-4 + 3q-3 - q-2 + q-1

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

{...}

A2 (sl(3)) Invariant:

- q-32 - q-30 + q-28 + q-24 + q-22 - q-20 - q-16 + 2q-8 + q-6 + q-2

HOMFLY-PT Polynomial:

a2 + a2z2 + a4 + 2a4z2 - 2a6 - 2a6z2 - a6z4 + 2a8 + 2a8z2 - a10

Kauffman Polynomial:

- a2 + a2z2 + a3z3 + a4 - 3a4z2 + 2a4z4 - a5z + 3a5z3 - 3a5z5 + a5z7 + 2a6 - 9a6z2 + 13a6z4 - 9a6z6 + 2a6z8 - 2a7z + 8a7z3 - 5a7z5 - 2a7z7 + a7z9 + 2a8 - 12a8z2 + 25a8z4 - 19a8z6 + 4a8z8 - 4a9z + 13a9z3 - 7a9z5 - 2a9z7 + a9z9 + a10 - 7a10z2 + 14a10z4 - 10a10z6 + 2a10z8 - 3a11z + 7a11z3 - 5a11z5 + a11z7

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

{3, -7}

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

trqj r = -9 r = -8 r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 j = -1                   1 j = -3                 1 1 j = -5               2     j = -7             2 1     j = -9           2 2       j = -11         2 2         j = -13       2 2           j = -15     1 2             j = -17   1 2               j = -19   1                 j = -21 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, 1]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 1]]
Out[3]=	PD[X[4, 2, 5, 1], X[8, 3, 9, 4], X[10, 6, 11, 5], X[14, 7, 15, 8], > X[2, 9, 3, 10], X[11, 16, 12, 17], X[13, 20, 14, 21], X[6, 15, 7, 16], > X[17, 22, 18, 1], X[19, 12, 20, 13], X[21, 18, 22, 19]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 1]]
Out[4]=	GaussCode[1, -5, 2, -1, 3, -8, 4, -2, 5, -3, -6, 10, -7, -4, 8, 6, -9, 11, -10, > 7, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 1]]
Out[5]=	DTCode[4, 8, 10, 14, 2, -16, -20, 6, -22, -12, -18]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 1]][t]
Out[6]=	-2 7 2 -11 - t + - + 7 t - t t
In[7]:=	Conway[Knot[11, NonAlternating, 1]][z]
Out[7]=	2 4 1 + 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[9, 48], Knot[11, NonAlternating, 1]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 1]], KnotSignature[Knot[11, NonAlternating, 1]]}
Out[9]=	{27, -2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 1]][q]
Out[10]=	-10 2 3 4 4 4 4 3 -2 1 -q + -- - -- + -- - -- + -- - -- + -- - q + - 9 8 7 6 5 4 3 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, 1]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 1]][q]
Out[12]=	-32 -30 -28 -24 -22 -20 -16 2 -6 -2 -q - q + q + q + q - q - q + -- + q + q 8 q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 1]][a, z]
Out[13]=	2 4 6 8 10 2 2 4 2 6 2 8 2 6 4 a + a - 2 a + 2 a - a + a z + 2 a z - 2 a z + 2 a z - a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 1]][a, z]
Out[14]=	2 4 6 8 10 5 7 9 11 2 2 -a + a + 2 a + 2 a + a - a z - 2 a z - 4 a z - 3 a z + a z - 4 2 6 2 8 2 10 2 3 3 5 3 7 3 > 3 a z - 9 a z - 12 a z - 7 a z + a z + 3 a z + 8 a z + 9 3 11 3 4 4 6 4 8 4 10 4 5 5 > 13 a z + 7 a z + 2 a z + 13 a z + 25 a z + 14 a z - 3 a z - 7 5 9 5 11 5 6 6 8 6 10 6 5 7 > 5 a z - 7 a z - 5 a z - 9 a z - 19 a z - 10 a z + a z - 7 7 9 7 11 7 6 8 8 8 10 8 7 9 9 9 > 2 a z - 2 a z + a z + 2 a z + 4 a z + 2 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 1]], Vassiliev[3][Knot[11, NonAlternating, 1]]}
Out[15]=	{3, -7}
In[16]:=	Kh[Knot[11, NonAlternating, 1]][q, t]
Out[16]=	-3 1 1 1 1 2 1 2 2 q + - + ------ + ------ + ------ + ------ + ------ + ------ + ------ + q 21 9 19 8 17 8 17 7 15 7 15 6 13 6 q t q t q t q t q t q t q t 2 2 2 2 2 2 1 2 1 > ------ + ------ + ------ + ----- + ----- + ----- + ----- + ----- + ---- 13 5 11 5 11 4 9 4 9 3 7 3 7 2 5 2 3 q t 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 K11n1

K11a367 K11n2