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

K11n102 K11n104

Knotscape

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DrawMorseLink

PD Presentation:

X4251 X10,3,11,4 X14,5,15,6 X12,8,13,7 X9,20,10,21 X2,11,3,12 X16,13,17,14 X6,15,7,16 X17,22,18,1 X19,8,20,9 X21,18,22,19

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- t-3 + 7t-2 - 15t-1 + 19 - 15t + 7t2 - t3

Conway Polynomial:

1 + 4z2 + z4 - z6

Other knots with the same Alexander/Conway Polynomial:

{K11n10, K11n144, ...}

Determinant and Signature:

{65, -4}

Jones Polynomial:

- 2q-9 + 5q-8 - 8q-7 + 10q-6 - 11q-5 + 11q-4 - 8q-3 + 6q-2 - 3q-1 + 1

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

{K11n175, ...}

A2 (sl(3)) Invariant:

- 2q-28 + q-24 - 2q-22 + 2q-20 - q-18 + q-16 + 2q-14 - q-12 + 3q-10 - 2q-8 + q-6 + q-4 - q-2 + 1

HOMFLY-PT Polynomial:

a2 + 2a2z2 + a2z4 - a4 - 3a4z2 - 3a4z4 - a4z6 + 3a6 + 7a6z2 + 3a6z4 - 2a8 - 2a8z2

Kauffman Polynomial:

- a2 + 3a2z2 - 3a2z4 + a2z6 - a3z + 7a3z3 - 9a3z5 + 3a3z7 - a4 + 6a4z2 - 4a4z4 - 5a4z6 + 3a4z8 - 3a5z + 14a5z3 - 21a5z5 + 6a5z7 + a5z9 - 3a6 + 9a6z2 - 5a6z4 - 9a6z6 + 6a6z8 - a7z + 7a7z3 - 14a7z5 + 6a7z7 + a7z9 - 2a8 + 3a8z2 - 2a8z6 + 3a8z8 - a9z + 3a9z3 - 2a9z5 + 3a9z7 - 3a10z2 + 4a10z4 + a10z6 - 2a11z + 3a11z3

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

{4, -9}

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 11103. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.)

trqj r = -7 r = -6 r = -5 r = -4 r = -3 r = -2 r = -1 r = 0 r = 1 r = 2 j = 1                   1 j = -1                 2   j = -3               4 1   j = -5             5 3     j = -7           6 3       j = -9         5 5         j = -11       5 6           j = -13     3 5             j = -15   2 5               j = -17   3                 j = -19 2

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, 103]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 103]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[14, 5, 15, 6], X[12, 8, 13, 7], > X[9, 20, 10, 21], X[2, 11, 3, 12], X[16, 13, 17, 14], X[6, 15, 7, 16], > X[17, 22, 18, 1], X[19, 8, 20, 9], X[21, 18, 22, 19]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 103]]
Out[4]=	GaussCode[1, -6, 2, -1, 3, -8, 4, 10, -5, -2, 6, -4, 7, -3, 8, -7, -9, 11, -10, > 5, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 103]]
Out[5]=	DTCode[4, 10, 14, 12, -20, 2, 16, 6, -22, -8, -18]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 103]][t]
Out[6]=	-3 7 15 2 3 19 - t + -- - -- - 15 t + 7 t - t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 103]][z]
Out[7]=	2 4 6 1 + 4 z + z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[11, NonAlternating, 10], Knot[11, NonAlternating, 103], > Knot[11, NonAlternating, 144]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 103]], KnotSignature[Knot[11, NonAlternating, 103]]}
Out[9]=	{65, -4}
In[10]:=	J=Jones[Knot[11, NonAlternating, 103]][q]
Out[10]=	2 5 8 10 11 11 8 6 3 1 - -- + -- - -- + -- - -- + -- - -- + -- - - 9 8 7 6 5 4 3 2 q 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, 103], Knot[11, NonAlternating, 175]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 103]][q]
Out[12]=	2 -24 2 2 -18 -16 2 -12 3 2 -6 -4 1 - --- + q - --- + --- - q + q + --- - q + --- - -- + q + q - 28 22 20 14 10 8 q q q q q q -2 > q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 103]][a, z]
Out[13]=	2 4 6 8 2 2 4 2 6 2 8 2 2 4 a - a + 3 a - 2 a + 2 a z - 3 a z + 7 a z - 2 a z + a z - 4 4 6 4 4 6 > 3 a z + 3 a z - a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 103]][a, z]
Out[14]=	2 4 6 8 3 5 7 9 11 2 2 -a - a - 3 a - 2 a - a z - 3 a z - a z - a z - 2 a z + 3 a z + 4 2 6 2 8 2 10 2 3 3 5 3 7 3 > 6 a z + 9 a z + 3 a z - 3 a z + 7 a z + 14 a z + 7 a z + 9 3 11 3 2 4 4 4 6 4 10 4 3 5 > 3 a z + 3 a z - 3 a z - 4 a z - 5 a z + 4 a z - 9 a z - 5 5 7 5 9 5 2 6 4 6 6 6 8 6 > 21 a z - 14 a z - 2 a z + a z - 5 a z - 9 a z - 2 a z + 10 6 3 7 5 7 7 7 9 7 4 8 6 8 > a z + 3 a z + 6 a z + 6 a z + 3 a z + 3 a z + 6 a z + 8 8 5 9 7 9 > 3 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 103]], Vassiliev[3][Knot[11, NonAlternating, 103]]}
Out[15]=	{4, -9}
In[16]:=	Kh[Knot[11, NonAlternating, 103]][q, t]
Out[16]=	3 4 2 3 2 5 3 5 5 -- + -- + ------ + ------ + ------ + ------ + ------ + ------ + ------ + 5 3 19 7 17 6 15 6 15 5 13 5 13 4 11 4 q q q t q t q t q t q t q t q t 6 5 5 6 3 5 t 2 t 2 > ------ + ----- + ----- + ----- + ---- + ---- + -- + --- + q t 11 3 9 3 9 2 7 2 7 5 3 q q t q t q t q t q t q t q

Dror Bar-Natan: The Knot Atlas: 11 Crossing Knots: The Knot K11n103

K11n102 K11n104