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

K11n98 K11n100

Knotscape

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DrawMorseLink

PD Presentation:

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

Gauss Code:

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

DT (Dowker-Thistlethwaite) Code:

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

Alexander Polynomial:

- 3t-2 + 10t-1 - 13 + 10t - 3t2

Conway Polynomial:

1 - 2z2 - 3z4

Other knots with the same Alexander/Conway Polynomial:

{10144, ...}

Determinant and Signature:

{39, -2}

Jones Polynomial:

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

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

{...}

A2 (sl(3)) Invariant:

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

HOMFLY-PT Polynomial:

4a2 + 3a2z2 - 4a4 - 5a4z2 - 2a4z4 - a6z2 - a6z4 + a8 + a8z2

Kauffman Polynomial:

- 4a2 + 5a2z2 + 5a3z - 4a3z3 + 3a3z5 - 4a4 + 4a4z2 + 3a4z4 - 2a4z6 + a4z8 + 11a5z - 21a5z3 + 14a5z5 - 4a5z7 + a5z9 - 9a6z2 + 12a6z4 - 9a6z6 + 3a6z8 + 5a7z - 11a7z3 + 4a7z5 - 2a7z7 + a7z9 + a8 - 4a8z2 + 5a8z4 - 6a8z6 + 2a8z8 - a9z + 6a9z3 - 7a9z5 + 2a9z7 + 4a10z2 - 4a10z4 + a10z6

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

{-2, 6}

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 1199. 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                 3 j = -3               2 1 j = -5             4 2   j = -7           3 2     j = -9         3 4       j = -11       3 3         j = -13     1 3           j = -15   1 3             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, 99]]]
Out[2]=	-Graphics-
In[3]:=	PD[Knot[11, NonAlternating, 99]]
Out[3]=	PD[X[4, 2, 5, 1], X[10, 3, 11, 4], X[5, 14, 6, 15], X[7, 12, 8, 13], > X[9, 19, 10, 18], X[2, 11, 3, 12], X[13, 6, 14, 7], X[15, 20, 16, 21], > X[17, 22, 18, 1], X[19, 9, 20, 8], X[21, 16, 22, 17]]
In[4]:=	GaussCode[Knot[11, NonAlternating, 99]]
Out[4]=	GaussCode[1, -6, 2, -1, -3, 7, -4, 10, -5, -2, 6, 4, -7, 3, -8, 11, -9, 5, -10, > 8, -11, 9]
In[5]:=	DTCode[Knot[11, NonAlternating, 99]]
Out[5]=	DTCode[4, 10, -14, -12, -18, 2, -6, -20, -22, -8, -16]
In[6]:=	alex = Alexander[Knot[11, NonAlternating, 99]][t]
Out[6]=	3 10 2 -13 - -- + -- + 10 t - 3 t 2 t t
In[7]:=	Conway[Knot[11, NonAlternating, 99]][z]
Out[7]=	2 4 1 - 2 z - 3 z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 144], Knot[11, NonAlternating, 99]}
In[9]:=	{KnotDet[Knot[11, NonAlternating, 99]], KnotSignature[Knot[11, NonAlternating, 99]]}
Out[9]=	{39, -2}
In[10]:=	J=Jones[Knot[11, NonAlternating, 99]][q]
Out[10]=	-9 2 4 6 6 7 6 4 3 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, 99]}
In[12]:=	A2Invariant[Knot[11, NonAlternating, 99]][q]
Out[12]=	-28 2 -20 -16 3 -12 2 2 2 -4 3 q + --- - q - q - --- - q - --- + -- + -- + q + -- 22 14 10 8 6 2 q q q q q q
In[13]:=	HOMFLYPT[Knot[11, NonAlternating, 99]][a, z]
Out[13]=	2 4 8 2 2 4 2 6 2 8 2 4 4 6 4 4 a - 4 a + a + 3 a z - 5 a z - a z + a z - 2 a z - a z
In[14]:=	Kauffman[Knot[11, NonAlternating, 99]][a, z]
Out[14]=	2 4 8 3 5 7 9 2 2 4 2 -4 a - 4 a + a + 5 a z + 11 a z + 5 a z - a z + 5 a z + 4 a z - 6 2 8 2 10 2 3 3 5 3 7 3 9 3 > 9 a z - 4 a z + 4 a z - 4 a z - 21 a z - 11 a z + 6 a z + 4 4 6 4 8 4 10 4 3 5 5 5 7 5 > 3 a z + 12 a z + 5 a z - 4 a z + 3 a z + 14 a z + 4 a z - 9 5 4 6 6 6 8 6 10 6 5 7 7 7 > 7 a z - 2 a z - 9 a z - 6 a z + a z - 4 a z - 2 a z + 9 7 4 8 6 8 8 8 5 9 7 9 > 2 a z + a z + 3 a z + 2 a z + a z + a z
In[15]:=	{Vassiliev[2][Knot[11, NonAlternating, 99]], Vassiliev[3][Knot[11, NonAlternating, 99]]}
Out[15]=	{-2, 6}
In[16]:=	Kh[Knot[11, NonAlternating, 99]][q, t]
Out[16]=	-3 3 1 1 1 3 1 3 3 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 3 3 4 3 2 4 2 2 > ------ + ----- + ----- + ----- + ----- + ----- + ---- + ---- 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 K11n99

K11n98 K11n100