10 160

10_159

10_161

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10 160 Further Notes and Views

Knot presentations

Planar diagram presentation	X4251 X12,4,13,3 X7,14,8,15 X9,19,10,18 X19,7,20,6 X5,17,6,16 X17,11,18,10 X13,8,14,9 X15,1,16,20 X2,12,3,11
Gauss code	1, -10, 2, -1, -6, 5, -3, 8, -4, 7, 10, -2, -8, 3, -9, 6, -7, 4, -5, 9
Dowker-Thistlethwaite code	4 12 -16 -14 -18 2 -8 -20 -10 -6
Conway Notation	[-30:20:20]

Three dimensional invariants

Symmetry type Reversible Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 2 Maximal Thurston-Bennequin number [1][-10] Hyperbolic Volume 9.20392 A-Polynomial See Data:10 160/A-polynomial

[edit Notes for 10 160's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $2$ Topological 4 genus $2$ Concordance genus $3$ Rasmussen s-Invariant -4

[edit Notes for 10 160's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+4t^{2}-4t+3-4t^{-1}+4t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}-2z^{4}+3z^{2}+1$
2nd Alexander ideal (db, data sources)	$\{1\}$
Determinant and Signature	{ 21, 4 }
Jones polynomial	$-2q^{7}+3q^{6}-3q^{5}+4q^{4}-3q^{3}+3q^{2}-2q+1$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}a^{-4}+z^{4}a^{-2}-4z^{4}a^{-4}+z^{4}a^{-6}+3z^{2}a^{-2}-3z^{2}a^{-4}+3z^{2}a^{-6}+a^{-2}+a^{-6}-a^{-8}$
Kauffman polynomial (db, data sources)	$z^{8}a^{-4}+z^{8}a^{-6}+2z^{7}a^{-3}+3z^{7}a^{-5}+z^{7}a^{-7}+z^{6}a^{-2}-2z^{6}a^{-4}-3z^{6}a^{-6}-8z^{5}a^{-3}-11z^{5}a^{-5}-3z^{5}a^{-7}-4z^{4}a^{-2}-3z^{4}a^{-4}+2z^{4}a^{-6}+z^{4}a^{-8}+7z^{3}a^{-3}+10z^{3}a^{-5}+3z^{3}a^{-7}+4z^{2}a^{-2}+3z^{2}a^{-4}+z^{2}a^{-8}-za^{-3}-3za^{-5}+2za^{-9}-a^{-2}-a^{-6}-a^{-8}$
The A2 invariant	$1+2q^{-10}+2q^{-14}-q^{-22}-q^{-26}$
The G2 invariant	$q^{-2}-q^{-4}+3q^{-6}-4q^{-8}+3q^{-10}-4q^{-14}+10q^{-16}-8q^{-18}+7q^{-20}-q^{-22}-6q^{-24}+9q^{-26}-8q^{-28}+q^{-30}+5q^{-32}-8q^{-34}+6q^{-36}-7q^{-40}+13q^{-42}-12q^{-44}+5q^{-46}+2q^{-48}-7q^{-50}+12q^{-52}-8q^{-54}+7q^{-56}-q^{-58}+2q^{-60}+4q^{-62}-6q^{-64}+5q^{-66}-2q^{-68}+q^{-70}+3q^{-72}-5q^{-74}+3q^{-76}+3q^{-78}-8q^{-80}+10q^{-82}-11q^{-84}+3q^{-86}+6q^{-88}-13q^{-90}+11q^{-92}-6q^{-94}+5q^{-98}-7q^{-100}+q^{-102}+q^{-104}-3q^{-106}+3q^{-108}-2q^{-110}-2q^{-112}+3q^{-114}-4q^{-116}+2q^{-118}+2q^{-120}-2q^{-122}+q^{-124}$

Further Quantum Invariants

Further quantum knot invariants for 10_160.

A1 Invariants.

Weight	Invariant
1	$q-q^{-1}+q^{-3}+q^{-7}+q^{-9}+q^{-13}-2q^{-15}$
2	$q^{6}-q^{4}-2q^{2}+3+2q^{-2}-3q^{-4}+3q^{-8}-2q^{-12}+2q^{-14}+2q^{-16}-2q^{-18}+q^{-20}+3q^{-22}-2q^{-24}-q^{-26}+2q^{-28}-4q^{-32}+2q^{-36}-2q^{-38}-q^{-40}+q^{-42}+q^{-44}$
3	$q^{15}-q^{13}-2q^{11}+4q^{7}+4q^{5}-4q^{3}-7q-q^{-1}+7q^{-3}+7q^{-5}-3q^{-7}-8q^{-9}-3q^{-11}+9q^{-13}+10q^{-15}-3q^{-17}-12q^{-19}-2q^{-21}+12q^{-23}+6q^{-25}-10q^{-27}-7q^{-29}+8q^{-31}+9q^{-33}-6q^{-35}-7q^{-37}+6q^{-39}+6q^{-41}-5q^{-43}-6q^{-45}+2q^{-47}+5q^{-49}+q^{-51}-8q^{-53}-8q^{-55}+3q^{-57}+13q^{-59}+q^{-61}-14q^{-63}-8q^{-65}+15q^{-67}+11q^{-69}-7q^{-71}-11q^{-73}+9q^{-77}+2q^{-79}-2q^{-81}-2q^{-83}$

A2 Invariants.

Weight	Invariant
1,0	$1+2q^{-10}+2q^{-14}-q^{-22}-q^{-26}$
1,1	$q^{4}-2q^{2}+6-12q^{-2}+17q^{-4}-20q^{-6}+24q^{-8}-18q^{-10}+11q^{-12}+4q^{-14}-14q^{-16}+26q^{-18}-34q^{-20}+38q^{-22}-32q^{-24}+32q^{-26}-21q^{-28}+16q^{-30}-4q^{-32}-8q^{-34}+13q^{-36}-26q^{-38}+24q^{-40}-24q^{-42}+18q^{-44}-8q^{-46}+6q^{-50}-7q^{-52}+2q^{-54}-4q^{-56}+4q^{-60}-2q^{-62}+2q^{-64}$
2,0	$q^{4}-1-q^{-2}+q^{-4}+2q^{-6}+2q^{-12}+2q^{-14}-q^{-16}-q^{-18}+q^{-22}+2q^{-24}+3q^{-28}+q^{-30}+3q^{-32}+q^{-34}-q^{-36}-q^{-38}-q^{-40}-2q^{-42}-4q^{-44}-2q^{-46}-q^{-48}+2q^{-50}-q^{-52}-q^{-54}+q^{-56}+q^{-58}+q^{-60}-q^{-62}+q^{-66}$

A3 Invariants.

Weight	Invariant
0,1,0	$1-q^{-2}+q^{-4}+q^{-6}-q^{-8}+2q^{-10}+q^{-14}+2q^{-16}+q^{-20}+q^{-22}+q^{-24}+2q^{-28}+2q^{-32}-q^{-34}-q^{-36}-q^{-38}-4q^{-40}-q^{-42}-q^{-44}-q^{-46}+q^{-48}+2q^{-50}$
1,0,0	$q^{-1}+q^{-5}-q^{-7}+q^{-9}+q^{-13}+q^{-15}+q^{-17}+q^{-19}+q^{-23}-q^{-25}+q^{-27}-q^{-29}-q^{-33}-q^{-35}$

A4 Invariants.

Weight	Invariant
0,1,0,0	$q^{-2}+q^{-8}+q^{-12}+q^{-14}+q^{-18}+3q^{-20}+q^{-22}+3q^{-26}+3q^{-28}+q^{-30}-2q^{-32}+4q^{-34}+2q^{-36}-2q^{-38}+q^{-40}+3q^{-42}-3q^{-44}-q^{-46}+q^{-48}-3q^{-50}-4q^{-52}-q^{-54}-4q^{-58}-2q^{-60}+2q^{-62}+q^{-64}-2q^{-66}+2q^{-68}+2q^{-70}$
1,0,0,0	$q^{-2}+q^{-6}+q^{-12}+q^{-16}+2q^{-20}+q^{-24}+q^{-28}+q^{-34}-q^{-36}-q^{-40}-q^{-42}-q^{-44}$

B2 Invariants.

Weight	Invariant
0,1	$1-q^{-2}+3q^{-4}-3q^{-6}+3q^{-8}-2q^{-10}+2q^{-12}-q^{-14}+2q^{-18}-3q^{-20}+5q^{-22}-5q^{-24}+6q^{-26}-4q^{-28}+4q^{-30}-2q^{-32}+q^{-34}+q^{-36}-q^{-38}+2q^{-40}-3q^{-42}+3q^{-44}-3q^{-46}+q^{-48}-2q^{-50}$
1,0	$q^{2}-q^{-2}-q^{-4}+2q^{-6}+2q^{-8}-2q^{-10}-2q^{-12}+2q^{-14}+3q^{-16}-3q^{-20}+4q^{-24}+2q^{-26}-2q^{-28}-q^{-30}+q^{-32}+2q^{-34}-q^{-38}+3q^{-42}+q^{-44}-q^{-46}-q^{-48}+2q^{-50}+3q^{-52}-q^{-54}-4q^{-56}+2q^{-60}-q^{-62}-4q^{-64}-2q^{-66}+q^{-68}+q^{-70}-2q^{-72}-2q^{-74}+q^{-76}+q^{-78}+2q^{-80}$

D4 Invariants.

Weight	Invariant
1,0,0,0	$q^{-2}-q^{-4}+2q^{-6}-2q^{-8}+4q^{-10}-2q^{-12}+2q^{-14}-q^{-16}+3q^{-18}+2q^{-24}-q^{-26}+4q^{-28}-2q^{-30}+4q^{-32}-4q^{-34}+5q^{-36}-3q^{-38}+4q^{-40}-2q^{-42}+3q^{-44}-q^{-46}+q^{-48}-3q^{-52}-4q^{-56}-4q^{-60}+2q^{-62}-2q^{-64}+q^{-66}+2q^{-70}$

G2 Invariants.

Weight

Invariant

1,0

$q^{-2}-q^{-4}+3q^{-6}-4q^{-8}+3q^{-10}-4q^{-14}+10q^{-16}-8q^{-18}+7q^{-20}-q^{-22}-6q^{-24}+9q^{-26}-8q^{-28}+q^{-30}+5q^{-32}-8q^{-34}+6q^{-36}-7q^{-40}+13q^{-42}-12q^{-44}+5q^{-46}+2q^{-48}-7q^{-50}+12q^{-52}-8q^{-54}+7q^{-56}-q^{-58}+2q^{-60}+4q^{-62}-6q^{-64}+5q^{-66}-2q^{-68}+q^{-70}+3q^{-72}-5q^{-74}+3q^{-76}+3q^{-78}-8q^{-80}+10q^{-82}-11q^{-84}+3q^{-86}+6q^{-88}-13q^{-90}+11q^{-92}-6q^{-94}+5q^{-98}-7q^{-100}+q^{-102}+q^{-104}-3q^{-106}+3q^{-108}-2q^{-110}-2q^{-112}+3q^{-114}-4q^{-116}+2q^{-118}+2q^{-120}-2q^{-122}+q^{-124}$

.

Computer Talk

The above data is available with the Mathematica package KnotTheory`, as shown in the (simulated) Mathematica session below. Your input (in red) is realistic; all else should have the same content as in a real mathematica session, but with different formatting. This Mathematica session is also available (albeit only for the knot 5_2) as the notebook PolynomialInvariantsSession.nb.

(The path below may be different on your system, and possibly also the KnotTheory` date)

In[1]:=

AppendTo[$Path, "C:/drorbn/projects/KAtlas/"]; << KnotTheory`

Loading KnotTheory` version of August 31, 2006, 11:25:27.5625. Read more at http://katlas.math.toronto.edu/wiki/KnotTheory.

In[3]:=

K = Knot["10 160"];

In[4]:=

Alexander[K][t]

KnotTheory::loading: Loading precomputed data in PD4Knots`.

Out[4]=

$-t^{3}+4t^{2}-4t+3-4t^{-1}+4t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

$-z^{6}-2z^{4}+3z^{2}+1$

In[6]:=

Alexander[K, 2][t]

KnotTheory::credits: The program Alexander[K, r] to compute Alexander ideals was written by Jana Archibald at the University of Toronto in the summer of 2005.

Out[6]=

$\{1\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 21, 4 }

In[8]:=

Jones[K][q]

KnotTheory::loading: Loading precomputed data in Jones4Knots`.

Out[8]=

$-2q^{7}+3q^{6}-3q^{5}+4q^{4}-3q^{3}+3q^{2}-2q+1$

In[9]:=

HOMFLYPT[K][a, z]

KnotTheory::credits: The HOMFLYPT program was written by Scott Morrison.

Out[9]=

$-z^{6}a^{-4}+z^{4}a^{-2}-4z^{4}a^{-4}+z^{4}a^{-6}+3z^{2}a^{-2}-3z^{2}a^{-4}+3z^{2}a^{-6}+a^{-2}+a^{-6}-a^{-8}$

In[10]:=

Kauffman[K][a, z]

KnotTheory::loading: Loading precomputed data in Kauffman4Knots`.

Out[10]=

$z^{8}a^{-4}+z^{8}a^{-6}+2z^{7}a^{-3}+3z^{7}a^{-5}+z^{7}a^{-7}+z^{6}a^{-2}-2z^{6}a^{-4}-3z^{6}a^{-6}-8z^{5}a^{-3}-11z^{5}a^{-5}-3z^{5}a^{-7}-4z^{4}a^{-2}-3z^{4}a^{-4}+2z^{4}a^{-6}+z^{4}a^{-8}+7z^{3}a^{-3}+10z^{3}a^{-5}+3z^{3}a^{-7}+4z^{2}a^{-2}+3z^{2}a^{-4}+z^{2}a^{-8}-za^{-3}-3za^{-5}+2za^{-9}-a^{-2}-a^{-6}-a^{-8}$

Vassiliev invariants

V2 and V3:

(3, 6)

V2,1 through V6,9:

V2,1 V3,1 V4,1 V4,2 V4,3 V5,1 V5,2 V5,3 V5,4 V6,1 V6,2 V6,3 V6,4 V6,5 V6,6 V6,7 V6,8 V6,9 $12$ $48$ $72$ $238$ $50$ $576$ $1248$ $224$ $240$ $288$ $1152$ $2856$ $600$ ${\frac {65311}{10}}$ $-{\frac {1742}{5}}$ ${\frac {16874}{5}}$ ${\frac {171}{2}}$ ${\frac {4991}{10}}$

V_2,1 through V_6,9 were provided by Petr Dunin-Barkowski <barkovs@itep.ru>, Andrey Smirnov <asmirnov@itep.ru>, and Alexei Sleptsov <sleptsov@itep.ru> and uploaded on October 2010 by User:Drorbn. Note that they are normalized differently than V₂ and V₃.

Khovanov Homology

The coefficients of the monomials $t^{r}q^{j}$ are shown, along with their alternating sums $\chi$ (fixed $j$, alternation over $r$). 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 10 160. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -2 -1 0 1 2 3 4 5 χ 15               2 -2 13             1   1 11           2 2   0 9         2 1     1 7       1 2       1 5     2 2         0 3   1 2           1 1   1             -1 -1 1               1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=3$ $i=5$ $r=-2$ ${\mathbb {Z} }$ $r=-1$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=0$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }^{2}$ $r=1$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=2$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=3$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$ $r=4$ ${\mathbb {Z} }^{2}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=5$ ${\mathbb {Z} }_{2}^{2}$ ${\mathbb {Z} }^{2}$

Computer Talk

Much of the above data can be recomputed by Mathematica using the package KnotTheory`. See A Sample KnotTheory` Session.

${\textrm {Include}}({\textrm {ColouredJonesM.mhtml}})$

In[1]:=	<< KnotTheory`
Loading KnotTheory` (version of August 17, 2005, 14:44:34)...
In[2]:=	Crossings[Knot[10, 160]]
Out[2]=	10
In[3]:=	PD[Knot[10, 160]]
Out[3]=	PD[X[4, 2, 5, 1], X[12, 4, 13, 3], X[7, 14, 8, 15], X[9, 19, 10, 18], X[19, 7, 20, 6], X[5, 17, 6, 16], X[17, 11, 18, 10], X[13, 8, 14, 9], X[15, 1, 16, 20], X[2, 12, 3, 11]]
In[4]:=	GaussCode[Knot[10, 160]]
Out[4]=	GaussCode[1, -10, 2, -1, -6, 5, -3, 8, -4, 7, 10, -2, -8, 3, -9, 6, -7, 4, -5, 9]
In[5]:=	BR[Knot[10, 160]]
Out[5]=	BR[4, {1, 1, 1, 2, 1, 1, -3, 2, -1, 2, -3}]
In[6]:=	alex = Alexander[Knot[10, 160]][t]
Out[6]=	-3 4 4 2 3 3 - t + -- - - - 4 t + 4 t - t 2 t t
In[7]:=	Conway[Knot[10, 160]][z]
Out[7]=	2 4 6 1 + 3 z - 2 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 160], Knot[11, NonAlternating, 118]}
In[9]:=	{KnotDet[Knot[10, 160]], KnotSignature[Knot[10, 160]]}
Out[9]=	{21, 4}
In[10]:=	J=Jones[Knot[10, 160]][q]
Out[10]=	2 3 4 5 6 7 1 - 2 q + 3 q - 3 q + 4 q - 3 q + 3 q - 2 q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 160]}
In[12]:=	A2Invariant[Knot[10, 160]][q]
Out[12]=	10 14 22 26 1 + 2 q + 2 q - q - q
In[13]:=	Kauffman[Knot[10, 160]][a, z]
Out[13]=	2 2 2 3 3 -8 -6 -2 2 z 3 z z z 3 z 4 z 3 z 10 z -a - a - a + --- - --- - -- + -- + ---- + ---- + ---- + ----- + 9 5 3 8 4 2 7 5 a a a a a a a a 3 4 4 4 4 5 5 5 6 6 7 z z 2 z 3 z 4 z 3 z 11 z 8 z 3 z 2 z ---- + -- + ---- - ---- - ---- - ---- - ----- - ---- - ---- - ---- + 3 8 6 4 2 7 5 3 6 4 a a a a a a a a a a 6 7 7 7 8 8 z z 3 z 2 z z z -- + -- + ---- + ---- + -- + -- 2 7 5 3 6 4 a a a a a a
In[14]:=	{Vassiliev[2][Knot[10, 160]], Vassiliev[3][Knot[10, 160]]}
Out[14]=	{0, 6}
In[15]:=	Kh[Knot[10, 160]][q, t]
Out[15]=	3 3 5 1 q q 5 7 7 2 9 2 2 q + 2 q + ---- + - + -- + 2 q t + q t + 2 q t + 2 q t + 2 t t q t 9 3 11 3 11 4 13 4 15 5 q t + 2 q t + 2 q t + q t + 2 q t