10 115

10_114

10_116

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10 115 Further Notes and Views

Knot presentations

Planar diagram presentation	X6271 X14,6,15,5 X20,15,1,16 X16,7,17,8 X8,19,9,20 X18,11,19,12 X10,4,11,3 X4,10,5,9 X12,17,13,18 X2,14,3,13
Gauss code	1, -10, 7, -8, 2, -1, 4, -5, 8, -7, 6, -9, 10, -2, 3, -4, 9, -6, 5, -3
Dowker-Thistlethwaite code	6 10 14 16 4 18 2 20 12 8
Conway Notation	[8*20.20]

Three dimensional invariants

Symmetry type Negative amphicheiral Unknotting number 2 3-genus 3 Bridge index 3 Super bridge index Missing Nakanishi index 1 Maximal Thurston-Bennequin number [-6][-6] Hyperbolic Volume 16.638 A-Polynomial See Data:10 115/A-polynomial

[edit Notes for 10 115's three dimensional invariants]

Four dimensional invariants

Smooth 4 genus $1$ Topological 4 genus $1$ Concordance genus $3$ Rasmussen s-Invariant 0

[edit Notes for 10 115's four dimensional invariants]

Polynomial invariants

Alexander polynomial	$-t^{3}+9t^{2}-26t+37-26t^{-1}+9t^{-2}-t^{-3}$
Conway polynomial	$-z^{6}+3z^{4}+z^{2}+1$
2nd Alexander ideal (db, data sources)	$\left\{2,t^{2}+t+1\right\}$
Determinant and Signature	{ 109, 0 }
Jones polynomial	$-q^{5}+4q^{4}-9q^{3}+14q^{2}-17q+19-17q^{-1}+14q^{-2}-9q^{-3}+4q^{-4}-q^{-5}$
HOMFLY-PT polynomial (db, data sources)	$-z^{6}+2a^{2}z^{4}+2z^{4}a^{-2}-z^{4}-a^{4}z^{2}+a^{2}z^{2}+z^{2}a^{-2}-z^{2}a^{-4}+z^{2}-a^{2}-a^{-2}+3$
Kauffman polynomial (db, data sources)	$3az^{9}+3z^{9}a^{-1}+8a^{2}z^{8}+8z^{8}a^{-2}+16z^{8}+8a^{3}z^{7}+13az^{7}+13z^{7}a^{-1}+8z^{7}a^{-3}+4a^{4}z^{6}-9a^{2}z^{6}-9z^{6}a^{-2}+4z^{6}a^{-4}-26z^{6}+a^{5}z^{5}-13a^{3}z^{5}-34az^{5}-34z^{5}a^{-1}-13z^{5}a^{-3}+z^{5}a^{-5}-5a^{4}z^{4}+a^{2}z^{4}+z^{4}a^{-2}-5z^{4}a^{-4}+12z^{4}-a^{5}z^{3}+8a^{3}z^{3}+22az^{3}+22z^{3}a^{-1}+8z^{3}a^{-3}-z^{3}a^{-5}+2a^{4}z^{2}-a^{2}z^{2}-z^{2}a^{-2}+2z^{2}a^{-4}-6z^{2}-2a^{3}z-5az-5za^{-1}-2za^{-3}+a^{2}+a^{-2}+3$
The A2 invariant	$-q^{16}+q^{14}+2q^{12}-4q^{10}+2q^{8}-q^{6}-2q^{4}+5q^{2}-1+5q^{-2}-2q^{-4}-q^{-6}+2q^{-8}-4q^{-10}+2q^{-12}+q^{-14}-q^{-16}$
The G2 invariant	$q^{80}-3q^{78}+7q^{76}-13q^{74}+16q^{72}-17q^{70}+8q^{68}+17q^{66}-53q^{64}+98q^{62}-130q^{60}+121q^{58}-62q^{56}-61q^{54}+225q^{52}-360q^{50}+410q^{48}-311q^{46}+62q^{44}+258q^{42}-536q^{40}+646q^{38}-522q^{36}+193q^{34}+206q^{32}-514q^{30}+589q^{28}-396q^{26}+28q^{24}+339q^{22}-530q^{20}+436q^{18}-110q^{16}-314q^{14}+652q^{12}-743q^{10}+555q^{8}-133q^{6}-361q^{4}+759q^{2}-907+759q^{-2}-361q^{-4}-133q^{-6}+555q^{-8}-743q^{-10}+652q^{-12}-314q^{-14}-110q^{-16}+436q^{-18}-530q^{-20}+339q^{-22}+28q^{-24}-396q^{-26}+589q^{-28}-514q^{-30}+206q^{-32}+193q^{-34}-522q^{-36}+646q^{-38}-536q^{-40}+258q^{-42}+62q^{-44}-311q^{-46}+410q^{-48}-360q^{-50}+225q^{-52}-61q^{-54}-62q^{-56}+121q^{-58}-130q^{-60}+98q^{-62}-53q^{-64}+17q^{-66}+8q^{-68}-17q^{-70}+16q^{-72}-13q^{-74}+7q^{-76}-3q^{-78}+q^{-80}$

Further Quantum Invariants

Further quantum knot invariants for 10_115.

A1 Invariants.

Weight	Invariant
1	$-q^{11}+3q^{9}-5q^{7}+5q^{5}-3q^{3}+2q+2q^{-1}-3q^{-3}+5q^{-5}-5q^{-7}+3q^{-9}-q^{-11}$
2	$q^{32}-3q^{30}+q^{28}+11q^{26}-18q^{24}-9q^{22}+44q^{20}-24q^{18}-43q^{16}+65q^{14}-64q^{10}+42q^{8}+27q^{6}-45q^{4}-2q^{2}+37-2q^{-2}-45q^{-4}+27q^{-6}+42q^{-8}-64q^{-10}+65q^{-14}-43q^{-16}-24q^{-18}+44q^{-20}-9q^{-22}-18q^{-24}+11q^{-26}+q^{-28}-3q^{-30}+q^{-32}$
3	$-q^{63}+3q^{61}-q^{59}-7q^{57}+2q^{55}+21q^{53}+4q^{51}-58q^{49}-25q^{47}+108q^{45}+93q^{43}-151q^{41}-226q^{39}+156q^{37}+389q^{35}-73q^{33}-540q^{31}-100q^{29}+629q^{27}+303q^{25}-602q^{23}-495q^{21}+480q^{19}+618q^{17}-298q^{15}-650q^{13}+107q^{11}+599q^{9}+81q^{7}-504q^{5}-239q^{3}+379q+379q^{-1}-239q^{-3}-504q^{-5}+81q^{-7}+599q^{-9}+107q^{-11}-650q^{-13}-298q^{-15}+618q^{-17}+480q^{-19}-495q^{-21}-602q^{-23}+303q^{-25}+629q^{-27}-100q^{-29}-540q^{-31}-73q^{-33}+389q^{-35}+156q^{-37}-226q^{-39}-151q^{-41}+93q^{-43}+108q^{-45}-25q^{-47}-58q^{-49}+4q^{-51}+21q^{-53}+2q^{-55}-7q^{-57}-q^{-59}+3q^{-61}-q^{-63}$

A2 Invariants.

Weight	Invariant
1,0	$-q^{16}+q^{14}+2q^{12}-4q^{10}+2q^{8}-q^{6}-2q^{4}+5q^{2}-1+5q^{-2}-2q^{-4}-q^{-6}+2q^{-8}-4q^{-10}+2q^{-12}+q^{-14}-q^{-16}$
2,0	$q^{42}-q^{40}-3q^{38}+2q^{36}+8q^{34}-17q^{30}-5q^{28}+24q^{26}+4q^{24}-27q^{22}-4q^{20}+33q^{18}+10q^{16}-38q^{14}-q^{12}+26q^{10}-9q^{8}-18q^{6}+9q^{4}+12q^{2}-6+12q^{-2}+9q^{-4}-18q^{-6}-9q^{-8}+26q^{-10}-q^{-12}-38q^{-14}+10q^{-16}+33q^{-18}-4q^{-20}-27q^{-22}+4q^{-24}+24q^{-26}-5q^{-28}-17q^{-30}+8q^{-34}+2q^{-36}-3q^{-38}-q^{-40}+q^{-42}$

A3 Invariants.

Weight	Invariant
0,1,0	$q^{34}-3q^{32}+q^{30}+8q^{28}-15q^{26}+3q^{24}+26q^{22}-35q^{20}+4q^{18}+40q^{16}-49q^{14}+2q^{12}+39q^{10}-35q^{8}-5q^{6}+26q^{4}-q^{2}-8-q^{-2}+26q^{-4}-5q^{-6}-35q^{-8}+39q^{-10}+2q^{-12}-49q^{-14}+40q^{-16}+4q^{-18}-35q^{-20}+26q^{-22}+3q^{-24}-15q^{-26}+8q^{-28}+q^{-30}-3q^{-32}+q^{-34}$
1,0,0	$-q^{21}+q^{19}+2q^{15}-4q^{13}+3q^{11}-4q^{9}+q^{7}-2q^{5}+4q^{3}+2q+2q^{-1}+4q^{-3}-2q^{-5}+q^{-7}-4q^{-9}+3q^{-11}-4q^{-13}+2q^{-15}+q^{-19}-q^{-21}$

B2 Invariants.

Weight	Invariant
0,1	$-q^{34}+3q^{32}-7q^{30}+14q^{28}-25q^{26}+37q^{24}-48q^{22}+57q^{20}-60q^{18}+54q^{16}-41q^{14}+20q^{12}+7q^{10}-37q^{8}+67q^{6}-90q^{4}+109q^{2}-114+109q^{-2}-90q^{-4}+67q^{-6}-37q^{-8}+7q^{-10}+20q^{-12}-41q^{-14}+54q^{-16}-60q^{-18}+57q^{-20}-48q^{-22}+37q^{-24}-25q^{-26}+14q^{-28}-7q^{-30}+3q^{-32}-q^{-34}$
1,0	$q^{56}-3q^{52}-3q^{50}+4q^{48}+11q^{46}+q^{44}-20q^{42}-16q^{40}+20q^{38}+37q^{36}-3q^{34}-51q^{32}-25q^{30}+47q^{28}+50q^{26}-25q^{24}-64q^{22}-5q^{20}+59q^{18}+26q^{16}-44q^{14}-37q^{12}+28q^{10}+40q^{8}-13q^{6}-40q^{4}+6q^{2}+43+6q^{-2}-40q^{-4}-13q^{-6}+40q^{-8}+28q^{-10}-37q^{-12}-44q^{-14}+26q^{-16}+59q^{-18}-5q^{-20}-64q^{-22}-25q^{-24}+50q^{-26}+47q^{-28}-25q^{-30}-51q^{-32}-3q^{-34}+37q^{-36}+20q^{-38}-16q^{-40}-20q^{-42}+q^{-44}+11q^{-46}+4q^{-48}-3q^{-50}-3q^{-52}+q^{-56}$

G2 Invariants.

Weight

Invariant

1,0

$q^{80}-3q^{78}+7q^{76}-13q^{74}+16q^{72}-17q^{70}+8q^{68}+17q^{66}-53q^{64}+98q^{62}-130q^{60}+121q^{58}-62q^{56}-61q^{54}+225q^{52}-360q^{50}+410q^{48}-311q^{46}+62q^{44}+258q^{42}-536q^{40}+646q^{38}-522q^{36}+193q^{34}+206q^{32}-514q^{30}+589q^{28}-396q^{26}+28q^{24}+339q^{22}-530q^{20}+436q^{18}-110q^{16}-314q^{14}+652q^{12}-743q^{10}+555q^{8}-133q^{6}-361q^{4}+759q^{2}-907+759q^{-2}-361q^{-4}-133q^{-6}+555q^{-8}-743q^{-10}+652q^{-12}-314q^{-14}-110q^{-16}+436q^{-18}-530q^{-20}+339q^{-22}+28q^{-24}-396q^{-26}+589q^{-28}-514q^{-30}+206q^{-32}+193q^{-34}-522q^{-36}+646q^{-38}-536q^{-40}+258q^{-42}+62q^{-44}-311q^{-46}+410q^{-48}-360q^{-50}+225q^{-52}-61q^{-54}-62q^{-56}+121q^{-58}-130q^{-60}+98q^{-62}-53q^{-64}+17q^{-66}+8q^{-68}-17q^{-70}+16q^{-72}-13q^{-74}+7q^{-76}-3q^{-78}+q^{-80}$

.

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["10 115"];

In[4]:=

Alexander[K][t]

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

Out[4]=

$-t^{3}+9t^{2}-26t+37-26t^{-1}+9t^{-2}-t^{-3}$

In[5]:=

Conway[K][z]

Out[5]=

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

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]=

$\left\{2,t^{2}+t+1\right\}$

In[7]:=

{KnotDet[K], KnotSignature[K]}

Out[7]=

{ 109, 0 }

In[8]:=

Jones[K][q]

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

Out[8]=

$-q^{5}+4q^{4}-9q^{3}+14q^{2}-17q+19-17q^{-1}+14q^{-2}-9q^{-3}+4q^{-4}-q^{-5}$

In[9]:=

HOMFLYPT[K][a, z]

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

Out[9]=

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

In[10]:=

Kauffman[K][a, z]

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

Out[10]=

$3az^{9}+3z^{9}a^{-1}+8a^{2}z^{8}+8z^{8}a^{-2}+16z^{8}+8a^{3}z^{7}+13az^{7}+13z^{7}a^{-1}+8z^{7}a^{-3}+4a^{4}z^{6}-9a^{2}z^{6}-9z^{6}a^{-2}+4z^{6}a^{-4}-26z^{6}+a^{5}z^{5}-13a^{3}z^{5}-34az^{5}-34z^{5}a^{-1}-13z^{5}a^{-3}+z^{5}a^{-5}-5a^{4}z^{4}+a^{2}z^{4}+z^{4}a^{-2}-5z^{4}a^{-4}+12z^{4}-a^{5}z^{3}+8a^{3}z^{3}+22az^{3}+22z^{3}a^{-1}+8z^{3}a^{-3}-z^{3}a^{-5}+2a^{4}z^{2}-a^{2}z^{2}-z^{2}a^{-2}+2z^{2}a^{-4}-6z^{2}-2a^{3}z-5az-5za^{-1}-2za^{-3}+a^{2}+a^{-2}+3$

Vassiliev invariants

V2 and V3:

(1, 0)

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 $4$ $0$ $8$ $-{\frac {34}{3}}$ $-{\frac {62}{3}}$ $0$ $0$ $0$ $0$ ${\frac {32}{3}}$ $0$ $-{\frac {136}{3}}$ $-{\frac {248}{3}}$ $-{\frac {1649}{30}}$ ${\frac {566}{5}}$ $-{\frac {9538}{45}}$ ${\frac {305}{18}}$ $-{\frac {2609}{30}}$

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=$0 is the signature of 10 115. Nonzero entries off the critical diagonals (if any exist) are highlighted in red.

\   r    \    j   \ -5 -4 -3 -2 -1 0 1 2 3 4 5 χ 11                     1 -1 9                   3   3 7                 6 1   -5 5               8 3     5 3             9 6       -3 1           10 8         2 -1         8 10           2 -3       6 9             -3 -5     3 8               5 -7   1 6                 -5 -9   3                   3 -11 1                     -1

Integral Khovanov Homology (db, data source)

$\dim {\mathcal {G}}_{2r+i}\operatorname {KH} _{\mathbb {Z} }^{r}$ $i=-1$ $i=1$ $r=-5$ ${\mathbb {Z} }$ $r=-4$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}$ ${\mathbb {Z} }$ $r=-3$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=-2$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=-1$ ${\mathbb {Z} }^{9}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=0$ ${\mathbb {Z} }^{10}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{10}$ $r=1$ ${\mathbb {Z} }^{8}\oplus {\mathbb {Z} }_{2}^{9}$ ${\mathbb {Z} }^{9}$ $r=2$ ${\mathbb {Z} }^{6}\oplus {\mathbb {Z} }_{2}^{8}$ ${\mathbb {Z} }^{8}$ $r=3$ ${\mathbb {Z} }^{3}\oplus {\mathbb {Z} }_{2}^{6}$ ${\mathbb {Z} }^{6}$ $r=4$ ${\mathbb {Z} }\oplus {\mathbb {Z} }_{2}^{3}$ ${\mathbb {Z} }^{3}$ $r=5$ ${\mathbb {Z} }_{2}$ ${\mathbb {Z} }$

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, 115]]
Out[2]=	10
In[3]:=	PD[Knot[10, 115]]
Out[3]=	PD[X[6, 2, 7, 1], X[14, 6, 15, 5], X[20, 15, 1, 16], X[16, 7, 17, 8], X[8, 19, 9, 20], X[18, 11, 19, 12], X[10, 4, 11, 3], X[4, 10, 5, 9], X[12, 17, 13, 18], X[2, 14, 3, 13]]
In[4]:=	GaussCode[Knot[10, 115]]
Out[4]=	GaussCode[1, -10, 7, -8, 2, -1, 4, -5, 8, -7, 6, -9, 10, -2, 3, -4, 9, -6, 5, -3]
In[5]:=	BR[Knot[10, 115]]
Out[5]=	BR[5, {1, -2, 1, 3, 2, 2, -4, -3, 2, -3, -3, -4}]
In[6]:=	alex = Alexander[Knot[10, 115]][t]
Out[6]=	-3 9 26 2 3 37 - t + -- - -- - 26 t + 9 t - t 2 t t
In[7]:=	Conway[Knot[10, 115]][z]
Out[7]=	2 4 6 1 + z + 3 z - z
In[8]:=	Select[AllKnots[], (alex === Alexander[#][t])&]
Out[8]=	{Knot[10, 115]}
In[9]:=	{KnotDet[Knot[10, 115]], KnotSignature[Knot[10, 115]]}
Out[9]=	{109, 0}
In[10]:=	J=Jones[Knot[10, 115]][q]
Out[10]=	-5 4 9 14 17 2 3 4 5 19 - q + -- - -- + -- - -- - 17 q + 14 q - 9 q + 4 q - q 4 3 2 q q q q
In[11]:=	Select[AllKnots[], (J === Jones[#][q] || (J /. q-> 1/q) === Jones[#][q])&]
Out[11]=	{Knot[10, 115]}
In[12]:=	A2Invariant[Knot[10, 115]][q]
Out[12]=	-16 -14 2 4 2 -6 2 5 2 4 6 -1 - q + q + --- - --- + -- - q - -- + -- + 5 q - 2 q - q + 12 10 8 4 2 q q q q q 8 10 12 14 16 2 q - 4 q + 2 q + q - q
In[13]:=	Kauffman[Knot[10, 115]][a, z]
Out[13]=	2 2 -2 2 2 z 5 z 3 2 2 z z 2 2 3 + a + a - --- - --- - 5 a z - 2 a z - 6 z + ---- - -- - a z + 3 a 4 2 a a a 3 3 3 4 2 z 8 z 22 z 3 3 3 5 3 4 2 a z - -- + ---- + ----- + 22 a z + 8 a z - a z + 12 z - 5 3 a a a 4 4 5 5 5 5 z z 2 4 4 4 z 13 z 34 z 5 ---- + -- + a z - 5 a z + -- - ----- - ----- - 34 a z - 4 2 5 3 a a a a a 6 6 7 3 5 5 5 6 4 z 9 z 2 6 4 6 8 z 13 a z + a z - 26 z + ---- - ---- - 9 a z + 4 a z + ---- + 4 2 3 a a a 7 8 9 13 z 7 3 7 8 8 z 2 8 3 z 9 ----- + 13 a z + 8 a z + 16 z + ---- + 8 a z + ---- + 3 a z a 2 a a
In[14]:=	{Vassiliev[2][Knot[10, 115]], Vassiliev[3][Knot[10, 115]]}
Out[14]=	{0, 0}
In[15]:=	Kh[Knot[10, 115]][q, t]
Out[15]=	10 1 3 1 6 3 8 6 -- + 10 q + ------ + ----- + ----- + ----- + ----- + ----- + ----- + q 11 5 9 4 7 4 7 3 5 3 5 2 3 2 q t q t q t q t q t q t q t 9 8 3 3 2 5 2 5 3 7 3 ---- + --- + 8 q t + 9 q t + 6 q t + 8 q t + 3 q t + 6 q t + 3 q t q t 7 4 9 4 11 5 q t + 3 q t + q t